Simultaneous linear equations

1. Tables & graphs

Equations and solutions

The equation y=32x+1 is represented with the diagram below. However, there is a number missing from the y values.

  1. Compute the value of y.
    Answer: y=
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]
    Substitute the correct input value into the equation and calculate the corresponding value of y.
    STEP: <no title>
    [−2 points ⇒ 0 / 2 points left]

    This question asks us to find the y value for the equation y=32x+1 when x=8.

    Let the equation show you the answer: substitute x=8 into the equation and calculate the missing value, y.

    y=32x+1y=32(8)+1y=12+1y=11

    The missing value from the diagram is the output value of y=11.


    Submit your answer as:
  2. Is (2;6) a solution to the equation y=32x+1?
    Answer: The correct solution is:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    You must test the numbers in the equation to see if the numbers agree with the equation or not!
    STEP: <no title>
    [−1 point ⇒ 0 / 1 points left]

    You have an equation, and you must decide if (2;6) solves the equation. Remember that a solution for the equation has values for x and y which "agree with" the equation. If you substitute x=2 into the equation and the answer is y=6 then (2;6) is a solution to the equation; otherwise it is not.

    The only way to know is to give it a try:

    y=32x+1y=32(2)+1y=3+1y=2

    When you substitute x=2 into the equation, the output value is y=2. These numbers make the ordered pair (2;2), which does not agree with the ordered pair in the question. That means that the ordered pair given in the question does not agree with the equation. Thus (2;6) is not a solution to the equation y=32x+1.

    The answer from the list is: No, it is not a solution.


    Submit your answer as:

Filling in a table of values

Answer the questions which follow about the equation y=3x+5.

  1. Fill in the table of values below for the equation. Give your answers as fractions (no rounded decimals).
    Answer:
    x 9 8 3 2 10
    y 22 19 4
    fraction
    fraction
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]
    Find the correct x-value for each of the y-values that you need, and let the equation find the answer for you.
    STEP: <no title>
    [−2 points ⇒ 2 / 4 points left]

    There are two values missing from the table. Both of them are y-values, so you can find the corresponding x-values from the table and calculate the answers using the equation. Remember that the equation encodes all of the information about the relationship between x and y.

    For the first blank space in the table, substitute x=2 into the equation and calculate the missing value:

    y=3x+5y=3(2)+5y=6+5y=1

    The first missing value from the table is y=1.


    STEP: <no title>
    [−2 points ⇒ 0 / 4 points left]

    Repeat the process for the first answer, but now with x=10. (We are not showing this calcualtion here, only the result.)

    The second missing value from the table is y=35.


    Submit your answer as: and
  2. How many solutions are there for the equation?
    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    The table of values above shows 5 ordered pairs which solve the equation. However, these are not the only ordered pairs which can solve the equation.
    STEP: <no title>
    [−1 point ⇒ 0 / 1 points left]

    The questions asks how many ways there are to solve the equation y=3x+5. Remember that a solution to the equation is an ordered pair which fits into the equation. In other words, a solution is a combination of x and y values which agree with the equation.

    The answer is that there are an infinite number of solutions to the equation! The table of values above shows 5 ordered pairs which solve the equation; however, there are many, many more! You can pick any number you want for x and the equation will tell you the value of y which belongs with it to get another ordered pair which solves the equation. If you keep changing the input value you put into the equation you will get more and more (and more and more...) solutions.

    The answer from the list is: There are an infinite number of solutions.


    Submit your answer as:

Inputs and outputs with a table of values

The diagram here is a "table of values." It shows input and output values for the equation y=2x+1.

Answer:
INSTRUCTION: If there is more than one correct answer for either question, give only one of the answers.
  1. What is the output for the input value of 7?
  2. Which input led to the output value of 3?
string
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Each number in the blue column is connected to the number next to it in the light brown column
STEP: Find the output value of 7
[−1 point ⇒ 1 / 2 points left]

A table of values lists two groups of numbers: the "input values" are in the left column and the "output values" are in the right column. The numbers are connected to each other by the equation, which defines the relationship between x and y.

x is always the input value for an equation, and y is always the output value. For the first question, we need to find the output value which corresponds to the input value of x=7: to find the corresponding output value, we substitute x=7 into the equation.

y=2x+1x=7plug iny=2(7)+1y=14+1y=15

The table of values shows the input-output pair x=7 and y=15:

The output value for the input of x=7 is 15.


STEP: Find the input value which gives an output of 3
[−1 point ⇒ 0 / 2 points left]

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of y=3 is 1.


Submit your answer as: and

Simultaneous equations on the Cartesian plane

The graph below shows these two equations:

y=x3y=2x+3

What is the solution to these two equations when they are solved simultaneously? Use the graph to answer the question.

Answer:

The solution to the equations is x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You need to find the coordinates of the point where the lines intersect.


STEP: Read the answer from the point of intersection
[−2 points ⇒ 0 / 2 points left]

This question is about two equations. We need to find the values of x and y which solve the equations 'simultaneously'. This means we want one pair of x and y values which solves both equations.

There are different methods to solve equations simultaneously. In this case we can use the graph: the solutions we want are the coordinates where the lines intersect. The intersection point is very special - it is the only point shared by both lines. And the coordinates of that point are the only numbers which solve both equations! In this case the lines intersect at the point (2;1).

We can prove that this is correct by substituting the values into each of the equations:

y=x3(1)=(2)31=1

and

y=2x+3(1)=2(2)+31=1

Perfect - the values x=2 and y=1 solve the equations simultaneously.

The correct answers are x=2 and y=1.


Submit your answer as: and

Inputs and outputs with a table of values

The diagram here is a "table of values." It shows input and output values for the equation y=2x1.

Answer:
INSTRUCTION: If there is more than one correct answer for either question, give only one of the answers.
  1. What is the output for the input value of 4?
  2. Which input led to the output value of 3?
numeric
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Each number in the blue column is connected to the number next to it in the light brown column
STEP: Find the output value of 4
[−1 point ⇒ 1 / 2 points left]

A table of values lists two groups of numbers: the "input values" are in the left column and the "output values" are in the right column. The numbers are connected to each other by the equation, which defines the relationship between x and y.

x is always the input value for an equation, and y is always the output value. For the first question, we need to find the output value which corresponds to the input value of x=4: to find the corresponding output value, we substitute x=4 into the equation.

y=2x1x=4plug iny=2(4)1y=81y=7

The table of values shows the input-output pair x=4 and y=7:

The output value for the input of x=4 is 7.


STEP: Find the input value which gives an output of 3
[−1 point ⇒ 0 / 2 points left]

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of y=3 is 2.


Submit your answer as: and

Exercises

Equations and solutions

The equation y=12x+4 is represented with the diagram below. However, there is a number missing from the y values.

  1. Determine the value of y.
    Answer: y=
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]
    Substitute the correct input value into the equation and calculate the corresponding value of y.
    STEP: <no title>
    [−2 points ⇒ 0 / 2 points left]

    This question asks us to find the y value for the equation y=12x+4 when x=3.

    Let the equation show you the answer: substitute x=3 into the equation and calculate the missing value, y.

    y=12x+4y=12(3)+4y=32+4y=112

    The missing value from the diagram is the output value of y=112.


    Submit your answer as:
  2. Is (4;2) a solution to the equation y=12x+4?
    Answer: The correct solution is:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    You must test the numbers in the equation to see if the numbers agree with the equation or not!
    STEP: <no title>
    [−1 point ⇒ 0 / 1 points left]

    You have an equation, and you must decide if (4;2) solves the equation. Remember that a solution for the equation has values for x and y which "agree with" the equation. If you substitute x=4 into the equation and the answer is y=2 then (4;2) is a solution to the equation; otherwise it is not.

    The only way to know is to give it a try:

    y=12x+4y=12(4)+4y=2+4y=2

    When you substitute x=4 into the equation, the output value is y=2. This is exactly what the ordered pair said it should be: the ordered pair agrees with the equation! Therefore the ordered pair (4;2) is a solution to the equation y=12x+4.

    The answer from the list is: Yes, it is a solution.


    Submit your answer as:

Equations and solutions

The equation y=32x4 is represented with the diagram below. However, there is a number missing from the y values.

  1. Determine the value of y.
    Answer: y=
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]
    Substitute the correct input value into the equation and calculate the corresponding value of y.
    STEP: <no title>
    [−2 points ⇒ 0 / 2 points left]

    This question asks us to find the y value for the equation y=32x4 when x=6.

    Let the equation show you the answer: substitute x=6 into the equation and calculate the missing value, y.

    y=32x4y=32(6)4y=94y=5

    The missing value from the diagram is the output value of y=5.


    Submit your answer as:
  2. Is (0;5) a solution to the equation y=32x4?
    Answer: The correct solution is:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    You must test the numbers in the equation to see if the numbers agree with the equation or not!
    STEP: <no title>
    [−1 point ⇒ 0 / 1 points left]

    You have an equation, and you must decide if (0;5) solves the equation. Remember that a solution for the equation has values for x and y which "agree with" the equation. If you substitute x=0 into the equation and the answer is y=5 then (0;5) is a solution to the equation; otherwise it is not.

    The only way to know is to give it a try:

    y=32x4y=32(0)4y=04y=4

    When you substitute x=0 into the equation, the output value is y=4. These numbers make the ordered pair (0;4), which does not agree with the ordered pair in the question. That means that the ordered pair given in the question does not agree with the equation. Thus (0;5) is not a solution to the equation y=32x4.

    The answer from the list is: No, it is not a solution.


    Submit your answer as:

Equations and solutions

The equation y=13x+4 is represented with the diagram below. However, there is a number missing from the y values.

  1. Find the value of y.
    Answer: y=
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]
    Substitute the correct input value into the equation and calculate the corresponding value of y.
    STEP: <no title>
    [−2 points ⇒ 0 / 2 points left]

    This question asks us to find the y value for the equation y=13x+4 when x=8.

    Let the equation show you the answer: substitute x=8 into the equation and calculate the missing value, y.

    y=13x+4y=13(8)+4y=83+4y=203

    The missing value from the diagram is the output value of y=203.


    Submit your answer as:
  2. Is (9;0) a solution to the equation y=13x+4?
    Answer: The correct solution is:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    You must test the numbers in the equation to see if the numbers agree with the equation or not!
    STEP: <no title>
    [−1 point ⇒ 0 / 1 points left]

    You have an equation, and you must decide if (9;0) solves the equation. Remember that a solution for the equation has values for x and y which "agree with" the equation. If you substitute x=9 into the equation and the answer is y=0 then (9;0) is a solution to the equation; otherwise it is not.

    The only way to know is to give it a try:

    y=13x+4y=13(9)+4y=3+4y=1

    When you substitute x=9 into the equation, the output value is y=1. These numbers make the ordered pair (9;1), which does not agree with the ordered pair in the question. That means that the ordered pair given in the question does not agree with the equation. Thus (9;0) is not a solution to the equation y=13x+4.

    The answer from the list is: No, it is not a solution.


    Submit your answer as:

Filling in a table of values

Answer the questions which follow about the equation y=3x43.

  1. Fill in the table of values below for the equation. Give your answers as fractions (no rounded decimals).
    Answer:
    x 3 1 6 7
    y 214 94
    fraction
    fraction
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]
    Find the correct x-value for each of the y-values that you need, and let the equation find the answer for you.
    STEP: <no title>
    [−2 points ⇒ 2 / 4 points left]

    There are two values missing from the table. Both of them are y-values, so you can find the corresponding x-values from the table and calculate the answers using the equation. Remember that the equation encodes all of the information about the relationship between x and y.

    For the first blank space in the table, substitute x=6 into the equation and calculate the missing value:

    y=3x43y=34(6)3y=923y=32

    The first missing value from the table is y=32.


    STEP: <no title>
    [−2 points ⇒ 0 / 4 points left]

    Repeat the process for the first answer, but now with x=7. (We are not showing this calcualtion here, only the result.)

    The second missing value from the table is y=94.


    Submit your answer as: and
  2. How many solutions are there for the equation?
    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    The table of values above shows 4 ordered pairs which solve the equation. However, these are not the only ordered pairs which can solve the equation.
    STEP: <no title>
    [−1 point ⇒ 0 / 1 points left]

    The questions asks how many ways there are to solve the equation y=3x43. Remember that a solution to the equation is an ordered pair which fits into the equation. In other words, a solution is a combination of x and y values which agree with the equation.

    The answer is that there are an infinite number of solutions to the equation! The table of values above shows 4 ordered pairs which solve the equation; however, there are many, many more! You can pick any number you want for x and the equation will tell you the value of y which belongs with it to get another ordered pair which solves the equation. If you keep changing the input value you put into the equation you will get more and more (and more and more...) solutions.

    The answer from the list is: There are an infinite number of solutions.


    Submit your answer as:

Filling in a table of values

Answer the questions which follow about the equation y=x23.

  1. Complete the table of values below for the equation. Give your answers as fractions (no rounded decimals).
    Answer:
    x 10 1 2 4
    y 72 1
    fraction
    fraction
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]
    Find the correct x-value for each of the y-values that you need, and let the equation find the answer for you.
    STEP: <no title>
    [−2 points ⇒ 2 / 4 points left]

    There are two values missing from the table. Both of them are y-values, so you can find the corresponding x-values from the table and calculate the answers using the equation. Remember that the equation encodes all of the information about the relationship between x and y.

    For the first blank space in the table, substitute x=10 into the equation and calculate the missing value:

    y=x23y=12(10)3y=53y=8

    The first missing value from the table is y=8.


    STEP: <no title>
    [−2 points ⇒ 0 / 4 points left]

    Repeat the process for the first answer, but now with x=2. (We are not showing this calcualtion here, only the result.)

    The second missing value from the table is y=2.


    Submit your answer as: and
  2. How many solutions are there for the equation?
    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    The table of values above shows 4 ordered pairs which solve the equation. However, these are not the only ordered pairs which can solve the equation.
    STEP: <no title>
    [−1 point ⇒ 0 / 1 points left]

    The questions asks how many ways there are to solve the equation y=x23. Remember that a solution to the equation is an ordered pair which fits into the equation. In other words, a solution is a combination of x and y values which agree with the equation.

    The answer is that there are an infinite number of solutions to the equation! The table of values above shows 4 ordered pairs which solve the equation; however, there are many, many more! You can pick any number you want for x and the equation will tell you the value of y which belongs with it to get another ordered pair which solves the equation. If you keep changing the input value you put into the equation you will get more and more (and more and more...) solutions.

    The answer from the list is: The number of solutions is ∞.


    Submit your answer as:

Filling in a table of values

Answer the questions which follow about the equation y=x3.

  1. Complete the table of values below for the equation. Give your answers as fractions (no rounded decimals).
    Answer:
    x 7 6 2 0
    y 73 2
    fraction
    fraction
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]
    Find the correct x-value for each of the y-values that you need, and let the equation find the answer for you.
    STEP: <no title>
    [−2 points ⇒ 2 / 4 points left]

    There are two values missing from the table. Both of them are y-values, so you can find the corresponding x-values from the table and calculate the answers using the equation. Remember that the equation encodes all of the information about the relationship between x and y.

    For the first blank space in the table, substitute x=2 into the equation and calculate the missing value:

    y=x3y=13(2)y=23

    The first missing value from the table is y=23.


    STEP: <no title>
    [−2 points ⇒ 0 / 4 points left]

    Repeat the process for the first answer, but now with x=0. (We are not showing this calcualtion here, only the result.)

    The second missing value from the table is y=0.


    Submit your answer as: and
  2. How many solutions are there for the equation?
    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    The table of values above shows 4 ordered pairs which solve the equation. However, these are not the only ordered pairs which can solve the equation.
    STEP: <no title>
    [−1 point ⇒ 0 / 1 points left]

    The questions asks how many ways there are to solve the equation y=x3. Remember that a solution to the equation is an ordered pair which fits into the equation. In other words, a solution is a combination of x and y values which agree with the equation.

    The answer is that there are an infinite number of solutions to the equation! The table of values above shows 4 ordered pairs which solve the equation; however, there are many, many more! You can pick any number you want for x and the equation will tell you the value of y which belongs with it to get another ordered pair which solves the equation. If you keep changing the input value you put into the equation you will get more and more (and more and more...) solutions.

    The answer from the list is: The number of solutions is ∞.


    Submit your answer as:

Inputs and outputs with a table of values

The diagram here is a "table of values." It shows input and output values for the equation y=4x+2.

Answer:
INSTRUCTION: If there is more than one correct answer for either question, give only one of the answers.
  1. What is the output for the input value of 8?
  2. Which input led to the output value of 10?
string
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Each of the input values, in the blue row, is connected to the output value below it in the light brown row.
STEP: Find the output value of 8
[−1 point ⇒ 1 / 2 points left]

A table of values lists two groups of numbers: this table lists the "input values" the top and the "output values" on the bottom. The numbers which are above and below each other are related to each other by the equation!

x is always the input value for an equation, and y is always the output value. For the first question, we need to find the output value which corresponds to the input value of x=8: to find the corresponding output value, we substitute x=8 into the equation.

y=4x+2x=8plug iny=4(8)+2y=32+2y=34

The table of values shows the input-output pair x=8 and y=34:

The output value for the input of x=8 is 34.


STEP: Find the input value which gives an output of 10
[−1 point ⇒ 0 / 2 points left]

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of y=10 is 2.


Submit your answer as: and

Inputs and outputs with a table of values

The diagram here is a "table of values." It shows input and output values for the equation y=x+2.

Answer:
INSTRUCTION: If there is more than one correct answer for either question, give only one of the answers.
  1. What is the output for the input value of 7?
  2. Which input led to the output value of 11?
string
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Each of the input values, in the blue row, is connected to the output value below it in the light brown row.
STEP: Find the output value of 7
[−1 point ⇒ 1 / 2 points left]

A table of values lists two groups of numbers: this table lists the "input values" the top and the "output values" on the bottom. The numbers which are above and below each other are related to each other by the equation!

x is always the input value for an equation, and y is always the output value. For the first question, we need to find the output value which corresponds to the input value of x=7: to find the corresponding output value, we substitute x=7 into the equation.

y=x+2x=7plug iny=(7)+2y=7+2y=9

The table of values shows the input-output pair x=7 and y=9:

The output value for the input of x=7 is 9.


STEP: Find the input value which gives an output of 11
[−1 point ⇒ 0 / 2 points left]

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of y=11 is 9.


Submit your answer as: and

Inputs and outputs with a table of values

The diagram here is a "table of values." It shows input and output values for the equation y=x+5.

Answer:
INSTRUCTION: If there is more than one correct answer for either question, give only one of the answers.
  1. What is the output for the input value of 9?
  2. Which input led to the output value of 15?
string
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Each number in the blue column is connected to the number next to it in the light brown column
STEP: Find the output value of 9
[−1 point ⇒ 1 / 2 points left]

A table of values lists two groups of numbers: the "input values" are in the left column and the "output values" are in the right column. The numbers are connected to each other by the equation, which defines the relationship between x and y.

x is always the input value for an equation, and y is always the output value. For the first question, we need to find the output value which corresponds to the input value of x=9: to find the corresponding output value, we substitute x=9 into the equation.

y=x+5x=9plug iny=(9)+5y=9+5y=14

The table of values shows the input-output pair x=9 and y=14:

The output value for the input of x=9 is 14.


STEP: Find the input value which gives an output of 15
[−1 point ⇒ 0 / 2 points left]

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of y=15 is 10.


Submit your answer as: and

Simultaneous equations on the Cartesian plane

The graph below shows these two equations:

y=2x5y=x2

What is the solution to these two equations when they are solved simultaneously? Use the graph to answer the question.

Answer:

The solution to the equations is x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You need to find the coordinates of the point where the lines intersect.


STEP: Read the answer from the point of intersection
[−2 points ⇒ 0 / 2 points left]

This question is about two equations. We need to find the values of x and y which solve the equations 'simultaneously'. This means we want one pair of x and y values which solves both equations.

There are different methods to solve equations simultaneously. In this case we can use the graph: the solutions we want are the coordinates where the lines intersect. The intersection point is very special - it is the only point shared by both lines. And the coordinates of that point are the only numbers which solve both equations! In this case the lines intersect at the point (2;1).

We can prove that this is correct by substituting the values into each of the equations:

y=2x5(1)=2(2)51=1

and

y=x2(1)=(2)21=1

Perfect - the values x=2 and y=1 solve the equations simultaneously.

The correct answers are x=2 and y=1.


Submit your answer as: and

Simultaneous equations on the Cartesian plane

The graph below shows these two equations:

y=3x+2y=x+4

What is the solution to these two equations when they are solved simultaneously? Use the graph to answer the question.

Answer:

The solution to the equations is x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You need to find the coordinates of the point where the lines intersect.


STEP: Read the answer from the point of intersection
[−2 points ⇒ 0 / 2 points left]

This question is about two equations. We need to find the values of x and y which solve the equations 'simultaneously'. This means we want one pair of x and y values which solves both equations.

There are different methods to solve equations simultaneously. In this case we can use the graph: the solutions we want are the coordinates where the lines intersect. The intersection point is very special - it is the only point shared by both lines. And the coordinates of that point are the only numbers which solve both equations! In this case the lines intersect at the point (1;5).

We can prove that this is correct by substituting the values into each of the equations:

y=3x+2(5)=3(1)+25=5

and

y=x+4(5)=(1)+45=5

Perfect - the values x=1 and y=5 solve the equations simultaneously.

The correct answers are x=1 and y=5.


Submit your answer as: and

Simultaneous equations on the Cartesian plane

The graph below shows these two equations:

y=3x23y=x2+5

What is the solution to these two equations when they are solved simultaneously? Use the graph to answer the question.

Answer:

The solution to the equations is x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You need to find the coordinates of the point where the lines intersect.


STEP: Read the answer from the point of intersection
[−2 points ⇒ 0 / 2 points left]

This question is about two equations. We need to find the values of x and y which solve the equations 'simultaneously'. This means we want one pair of x and y values which solves both equations.

There are different methods to solve equations simultaneously. In this case we can use the graph: the solutions we want are the coordinates where the lines intersect. The intersection point is very special - it is the only point shared by both lines. And the coordinates of that point are the only numbers which solve both equations! In this case the lines intersect at the point (4;3).

We can prove that this is correct by substituting the values into each of the equations:

y=3x23(3)=3(4)233=3

and

y=x2+5(3)=(4)2+53=3

Perfect - the values x=4 and y=3 solve the equations simultaneously.

The correct answers are x=4 and y=3.


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Inputs and outputs with a table of values

The diagram here is a "table of values." It shows input and output values for the equation y=2x5.

Answer:
INSTRUCTION: If there is more than one correct answer for either question, give only one of the answers.
  1. What is the output for the input value of 2?
  2. Which input led to the output value of 3?
numeric
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Each number in the blue column is connected to the number next to it in the light brown column
STEP: Find the output value of 2
[−1 point ⇒ 1 / 2 points left]

A table of values lists two groups of numbers: the "input values" are in the left column and the "output values" are in the right column. The numbers are connected to each other by the equation, which defines the relationship between x and y.

x is always the input value for an equation, and y is always the output value. For the first question, we need to find the output value which corresponds to the input value of x=2: to find the corresponding output value, we substitute x=2 into the equation.

y=2x5x=2plug iny=2(2)5y=45y=1

The table of values shows the input-output pair x=2 and y=1:

The output value for the input of x=2 is -1.


STEP: Find the input value which gives an output of 3
[−1 point ⇒ 0 / 2 points left]

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of y=3 is 4.


Submit your answer as: and

Inputs and outputs with a table of values

The diagram here is a "table of values." It shows input and output values for the equation y=2x1.

Answer:
INSTRUCTION: If there is more than one correct answer for either question, give only one of the answers.
  1. What is the output for the input value of 2?
  2. Which input led to the output value of 19?
numeric
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Each of the input values, in the blue row, is connected to the output value below it in the light brown row.
STEP: Find the output value of 2
[−1 point ⇒ 1 / 2 points left]

A table of values lists two groups of numbers: this table lists the "input values" the top and the "output values" on the bottom. The numbers which are above and below each other are related to each other by the equation!

x is always the input value for an equation, and y is always the output value. For the first question, we need to find the output value which corresponds to the input value of x=2: to find the corresponding output value, we substitute x=2 into the equation.

y=2x1x=2plug iny=2(2)1y=41y=3

The table of values shows the input-output pair x=2 and y=3:

The output value for the input of x=2 is 3.


STEP: Find the input value which gives an output of 19
[−1 point ⇒ 0 / 2 points left]

As with the first question, we must find the correct pair of numbers on the table. The input value which leads to an output of y=19 is 10.


Submit your answer as: and

Inputs and outputs with a table of values

The diagram here is a "table of values." It shows input and output values for the equation y=4.

Answer:
INSTRUCTION: If there is more than one correct answer for either question, give only one of the answers.
  1. What is the output for the input value of 6?
  2. Which input led to the output value of -4?
numeric
one-of
type(numeric.noerror)
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
Each number in the blue column is connected to the number next to it in the light brown column
STEP: Find the output value of 6
[−1 point ⇒ 1 / 2 points left]

A table of values lists two groups of numbers: the "input values" are in the left column and the "output values" are in the right column. The numbers are connected to each other by the equation, which defines the relationship between x and y.

x is always the input value for an equation, and y is always the output value. For the first question, we need to find the output value which corresponds to the input value of x=6: to find the corresponding output value, we substitute x=6 into the equation.

However, this equation does not even have an x in it - that is a bit weird! What this means is that the output is not related to the input: the output is always -4, no matter what the input is. (You can see that easily on the table of values: all of the output numbers are the same!) It is like a student who must wake up at 06:00 in the morning for school: if he slept early or stayed up late, it does not matter - he still must get up at 06:00. When he wakes does not depend on the time when he sleeps, just like the output for this equation does not depend on the input.

in x=6!can not plugy=4equal to 4y is always

The table of values shows the input-output pair x=6 and y=4:

The output value for the input of x=6 is -4.


STEP: Find the input value which gives an output of -4
[−1 point ⇒ 0 / 2 points left]

This question has multiple answers! Since the output value is always -4, all of the input values lead to the output value of y=4. Therefore, you can give any one of these answers for the second question: 1, 3, 5 or 6.


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2. Elimination method

Simultaneous equations: the elimination method

Here are two equations, which we can solve simultaneously using elimination:

5x+5y=205x2y=5

If we add these equations together, which of the terms will be eliminated (cancelled), and why does the elimination happen? Choose your answers from the choices below.

Answer:

We will eliminate because .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You should compare the coefficients in the two equations, and look for like terms: if you collect those terms, which terms will cancel?


STEP: Identify the terms which are easy to cancel
[−2 points ⇒ 0 / 2 points left]

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is elimination. The elimination method works by cancelling like terms which the two equations share. We need to figure out which of the terms in the equations will cancel, and why that cancellation happens.

In these two questions, there are two x-terms and two y-terms:

5x+5y=205x2y=5

The goal of elimination is to cancel out like terms by adding (or subtracting) the equations. That can be either the x-terms or the y-terms.

For the equations in this question, notice that the coefficients of x are equal and opposite. So if we add them together, they will cancel. So we will add the equations together to make that cancellation happen, like this:

5x+5y=205x2y=55x5x+5y2y=205disappear!x-terms 0x+3y=153y=15

The x-terms cancel each other out - that is what 'elimination' means! We get the 0x because the x-coefficients are equal and opposite: 5x+(5x) is equal to zero. So the elimination depends on which of the coefficients cancel out.

The correct answer choices are: We will eliminate the x-terms because the x-coefficients will cancel.


Submit your answer as: and

Solving simultaneous equations by elimination

Solve for m and n:

2m+15n=355m+3n=16
Answer: m= and n=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]
Look at the coefficients of the n-terms. They are not equal, but one of them is a multiple of the other; that is the key to using the elimination method to solve simultaneous equations.
STEP: Multiply one of the equations to set up the cancellation
[−2 points ⇒ 4 / 6 points left]

The first thing to do is compare the coefficients of the terms. We can see that the coefficient of the n-term in the first equation is a multiple of the second n-term: the 3 is 5 times as much as 15. If we multiply the entire second equation by this factor of 5 we will be able to cancel the n-terms between the equations:

5m+3n=16(5)5m+(5)3n=16(5)25m+15n=80
NOTE: We could also multiply the equation by −5. That will also make it possible to cancel the n-terms (it will just result in opposite signs for all of the coefficients).

STEP: Eliminate n by subtracting the equations
[−2 points ⇒ 2 / 6 points left]

Now eliminate the n-terms. Since the terms have the same signs, we must subtract one equation from the other to cancel those terms.

2m+15n=35to cancel the n-termsSubtract the equations(25m+15n)=8023m+0n=115

Now we can solve for m:

m=11523=5

STEP: Solve for n
[−2 points ⇒ 0 / 6 points left]

Finally, use the value of m to find the value of n. Remember that we can use either of the equations to do this, we can pick the easier equation. The second equation looks like a better choice because it has fewer negative signs than the first equation (in fact it does not have any!).

5m+3n=165(5)+3n=1625+3n=163n=9n=3

The answers to the pair of equations are m=5 and n=3.


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Solving simultaneous equations by elimination

Solve for x and y using the elimination method:

2x+y=183x+5y=27
Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]
Look at the coefficients of the y-terms. They are not equal, but one of them is a multiple of the other; that is the key to using the elimination method to solve simultaneous equations.
STEP: Start with the y-terms
[−2 points ⇒ 4 / 6 points left]

We need to solve the equations in this equation using elimination. (Substitution will also work, but the question says that we should use elimination.) As always for elimination, we need to use the two equations to cancel some of the terms.

By comparing the coefficients of the terms, we can see that the second y-term has a coefficient which is a multiple of the first y-term. In fact, 5 is 5 times as much as 1. We can make both coefficients of y the same by using this factor of 5 to multiply the entire first equation:

2x+y=18(5)2x+(5)y=18(5)10x+5y=90

STEP: Eliminate the y-variable and solve for x
[−2 points ⇒ 2 / 6 points left]

Now we can eliminate the y-terms by subtracting one equation from the other. Then we can solve for x.

10x+5y=90to cancel the y-terms:Subtract the equations(3x+5y)=(27)7x+0y=63

Now we can solve for x:

x=637=9

STEP: Substitute in for x
[−2 points ⇒ 0 / 6 points left]

The last step is to substitute y back into either of the equations so that we can find x. Here we will use the first equation:

2x+y=182(9)+y=1818+y=18y=0

The answers are x=9 and y=0.


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Setting up elimination: picking a multiple

A friend in your maths class started to solve these two equations simultaneously:

1=3k2p10=5k+5p

He is using the elimination method. He started by multiplying the first equation by 5, leading to this:

5=15k+10p10=5k+5p

But your friend is not sure what to do next, and he asks you for help. What number can you use to multiply the second equation so that the equations can be solved using elimination? Your answer should be an integer. Note that there may be more than one answer, but you should only give one answer.

Answer: You can multiply by .
one-of
type(numeric.noerror)
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Remember that the goal of elimination is to cancel either the k-terms or the p-terms. So you need to change one of the coefficients in the second equations so that you can cancel terms.


STEP: Find a way to make two of the terms cancel
[−1 point ⇒ 0 / 1 points left]

For this question we need to multiply the second equation by a number so that we can solve the equations by elimination. In the original equations, none of the coefficients were ready to cancel (because none of them are equal). The first equation was already multiplied by 5. We need to change one of the coefficients in the second equation so that we can cancel terms.

Specifically, we need to multiply 5 to get 15 so we can cancel the k-terms, or multiply the 5 to get 10 so we can cancel the p-terms.

For these equations, both options work. The coefficients of the k-terms and the p-terms in the first equation are multiples of the coefficients in the other equation. If we multiply by 3 we can cancel the k-terms. But we could also multiply by 2 which would allow us to cancel the p-terms. Both are good choices. Here we will multiply the second equation by 3.

term in the equationmultiply each10(3)=5(3)k+5(3)p30=15k+15p

The coefficients we want to cancel are equal and opposite. So eliminating the terms requires adding both sides of the equation:

5=15k+10pequationadd this30=15k+15p530=15k15k+10p+15p25=0k+25p
NOTE: If we had multiplied by 3, the signs in the second equation would be changed. Then we would subtract the equations instead. That means that 3 is also an acceptable answer.

The correct answer can be any of these numbers: 3, 3, 2 or 2.


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Exercises

Simultaneous equations: the elimination method

Here are two equations, which we can solve simultaneously using elimination:

3x+5y=154x5y=20

If we add these equations together, which of the terms will be eliminated (cancelled), and why does the elimination happen? Choose your answers from the choices below.

Answer:

We will eliminate because .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You should compare the coefficients in the two equations, and look for like terms: if you collect those terms, which terms will cancel?


STEP: Identify the terms which are easy to cancel
[−2 points ⇒ 0 / 2 points left]

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is elimination. The elimination method works by cancelling like terms which the two equations share. We need to figure out which of the terms in the equations will cancel, and why that cancellation happens.

In these two questions, there are two x-terms and two y-terms:

3x+5y=154x5y=20

The goal of elimination is to cancel out like terms by adding (or subtracting) the equations. That can be either the x-terms or the y-terms.

For the equations in this question, notice that the coefficients of y are equal and opposite. So if we add them together, they will cancel. So we will add the equations together to make that cancellation happen, like this:

3x+5y=154x5y=203x4x+5y5y=15+20disappear!y-terms 7x+0y=357x=35

The y-terms cancel each other out - that is what 'elimination' means! We get the 0y because the y-coefficients are equal and opposite: 5y+(5y) is equal to zero. So the elimination depends on which of the coefficients cancel out.

The correct answer choices are: We will eliminate the y-terms because the y-coefficients will cancel.


Submit your answer as: and

Simultaneous equations: the elimination method

Here are two equations, which we can solve simultaneously using elimination:

20=5x5y17=5x+2y

If we add these equations together, which of the terms will be eliminated (cancelled), and why does the elimination happen? Choose your answers from the choices below.

Answer:

We will eliminate because .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You should compare the coefficients in the two equations, and look for like terms: if you collect those terms, which terms will cancel?


STEP: Identify the terms which are easy to cancel
[−2 points ⇒ 0 / 2 points left]

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is elimination. The elimination method works by cancelling like terms which the two equations share. We need to figure out which of the terms in the equations will cancel, and why that cancellation happens.

In these two questions, there are two x-terms and two y-terms:

20=5x5y17=5x+2y

The goal of elimination is to cancel out like terms by adding (or subtracting) the equations. That can be either the x-terms or the y-terms.

For the equations in this question, notice that the coefficients of x are equal and opposite. So if we add them together, they will cancel. So we will add the equations together to make that cancellation happen, like this:

20=5x5y17=5x+2y20+17=5x5x5y+2ydisappear!x-terms 3=0x3y3=3y

The x-terms cancel each other out - that is what 'elimination' means! We get the 0x because the x-coefficients are equal and opposite: 5x+(5x) is equal to zero. So the elimination depends on which of the coefficients cancel out.

The correct answer choices are: We will eliminate the x-terms because the x-coefficients will cancel.


Submit your answer as: and

Simultaneous equations: the elimination method

Here are two equations, which we can solve simultaneously using elimination:

16=3xy12=4x+y

If we add these equations together, which of the terms will be eliminated (cancelled), and why does the elimination happen? Choose your answers from the choices below.

Answer:

We will eliminate because .

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

You should compare the coefficients in the two equations, and look for like terms: if you collect those terms, which terms will cancel?


STEP: Identify the terms which are easy to cancel
[−2 points ⇒ 0 / 2 points left]

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is elimination. The elimination method works by cancelling like terms which the two equations share. We need to figure out which of the terms in the equations will cancel, and why that cancellation happens.

In these two questions, there are two x-terms and two y-terms:

16=3xy12=4x+y

The goal of elimination is to cancel out like terms by adding (or subtracting) the equations. That can be either the x-terms or the y-terms.

For the equations in this question, notice that the coefficients of y are equal and opposite. So if we add them together, they will cancel. So we will add the equations together to make that cancellation happen, like this:

16=3xy12=4x+y16+12=3x+4xy+ydisappear!y-terms 28=7x+0y28=7x

The y-terms cancel each other out - that is what 'elimination' means! We get the 0y because the y-coefficients are equal and opposite: y+(y) is equal to zero. So the elimination depends on which of the coefficients cancel out.

The correct answer choices are: We will eliminate the y-terms because the y-coefficients will cancel.


Submit your answer as: and

Solving simultaneous equations by elimination

Solve for f and g:

5f8g=23f2g=4
Answer: f= and g=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]
Look at the coefficients of the g-terms. They are not equal, but one of them is a multiple of the other; that is the key to using the elimination method to solve simultaneous equations.
STEP: Multiply one of the equations to set up the cancellation
[−2 points ⇒ 4 / 6 points left]

The first thing to do is compare the coefficients of the terms. We can see that the coefficient of the g-term in the first equation is a multiple of the second g-term: the 2 is 4 times as much as 8. If we multiply the entire second equation by this factor of 4 we will be able to cancel the g-terms between the equations:

3f2g=4(4)3f(4)2g=4(4)12f8g=16
NOTE: We could also multiply the equation by −4. That will also make it possible to cancel the g-terms (it will just result in opposite signs for all of the coefficients).

STEP: Eliminate g by subtracting the equations
[−2 points ⇒ 2 / 6 points left]

Now eliminate the g-terms. Since the terms have the same signs, we must subtract one equation from the other to cancel those terms.

5f8g=2to cancel the g-termsSubtract the equations(12f8g)=(16)7f+0g=14

Now we can solve for f:

f=147=2

STEP: Solve for g
[−2 points ⇒ 0 / 6 points left]

Finally, use the value of f to find the value of g. Remember that we can use either of the equations to do this, we can pick the easier equation. The equations are pretty much the same (they have the same number of negative signs, for example, and neither has any fractions). So we will just pick the first equation.

5f8g=25(2)8g=2108g=28g=8g=1

The answers to the pair of equations are f=2 and g=1.


Submit your answer as: and

Solving simultaneous equations by elimination

Solve for a and b:

5a8b=163a+2b=38
Answer: a= and b=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]
Look at the coefficients of the b-terms. They are not equal, but one of them is a multiple of the other; that is the key to using the elimination method to solve simultaneous equations.
STEP: Multiply one of the equations to set up the cancellation
[−2 points ⇒ 4 / 6 points left]

The first thing to do is compare the coefficients of the terms. We can see that the coefficient of the b-term in the first equation is a multiple of the second b-term: the 2 is 4 times as much as 8. If we multiply the entire second equation by this factor of 4 we will be able to cancel the b-terms between the equations:

3a+2b=38(4)3a+(4)2b=38(4)12a+8b=152
NOTE: We could also multiply the equation by −4. That will also make it possible to cancel the b-terms (it will just result in opposite signs for all of the coefficients).

STEP: Eliminate b by adding the equations
[−2 points ⇒ 2 / 6 points left]

Now eliminate the b-terms. Since the terms have opposite signs, we must add one equation to the other to cancel those terms.

5a8b=16to cancel the b-termsAdd the equations+(12a+8b)=+15217a+0b=136

Now we can solve for a:

a=13617=8

STEP: Solve for b
[−2 points ⇒ 0 / 6 points left]

Finally, use the value of a to find the value of b. Remember that we can use either of the equations to do this, we can pick the easier equation. The second equation looks like a better choice because it has fewer negative signs than the first equation (in fact it does not have any!).

3a+2b=383(8)+2b=3824+2b=382b=14b=7

The answers to the pair of equations are a=8 and b=7.


Submit your answer as: and

Solving simultaneous equations by elimination

Solve for j and k:

12j+6k=66j+5k=11
Answer: j= and k=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]
Look at the coefficients of the j-terms. They are not equal, but one of them is a multiple of the other; that is the key to using the elimination method to solve simultaneous equations.
STEP: Multiply one of the equations to set up the cancellation
[−2 points ⇒ 4 / 6 points left]

The first thing to do is compare the coefficients of the terms. We can see that the coefficient of the j-term in the first equation is a multiple of the second j-term: the 6 is 2 times as much as 12. If we multiply the entire second equation by this factor of 2 we will be able to cancel the j-terms between the equations:

6j+5k=11(2)6j+(2)5k=11(2)12j+10k=22
NOTE: We could also multiply the equation by −2. That will also make it possible to cancel the j-terms (it will just result in opposite signs for all of the coefficients).

STEP: Eliminate j by subtracting the equations
[−2 points ⇒ 2 / 6 points left]

Now eliminate the j-terms. Since the terms have the same signs, we must subtract one equation from the other to cancel those terms.

12j+6k=6to cancel the j-termsSubtract the equations(12j+10k)=(22)0j4k=28

Now we can solve for k:

k=284=7

STEP: Solve for j
[−2 points ⇒ 0 / 6 points left]

Finally, use the value of k to find the value of j. Remember that we can use either of the equations to do this, we can pick the easier equation. In this case the first equation has fewer negatives than the second equation - in fact it has none. So we will use the first equation.

12j+6k=612j+6(7)=612j42=612j=48j=4

The answers to the pair of equations are j=4 and k=7.


Submit your answer as: and

Solving simultaneous equations by elimination

Solve for x and y using the elimination method:

x3y=23x5y=10
Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]
Look at the coefficients of the x-terms. They are not equal, but one of them is a multiple of the other; that is the key to using the elimination method to solve simultaneous equations.
STEP: Start with the x-terms
[−2 points ⇒ 4 / 6 points left]

We need to solve the equations in this equation using elimination. (Substitution will also work, but the question says that we should use elimination.) As always for elimination, we need to use the two equations to cancel some of the terms.

By comparing the coefficients of the terms, we can see that the second x-term has a coefficient which is a multiple of the first x-term. In fact, 3 is 3 times as much as 1. We can make both coefficients of x the same by using this factor of 3 to multiply the entire first equation:

x3y=2(3)x(3)3y=2(3)3x9y=6

STEP: Eliminate the x-variable and solve for y
[−2 points ⇒ 2 / 6 points left]

Now we can eliminate the x-terms by subtracting one equation from the other. Then we can solve for y.

3x9y=6to cancel the x-terms:Subtract the equations(3x5y)=(10)0x4y=16

Now we can solve for y:

y=164=4

STEP: Substitute in for y
[−2 points ⇒ 0 / 6 points left]

The last step is to substitute y back into either of the equations so that we can find x. Here we will use the first equation:

x3y=2x3(4)=2x12=2x=10

The answers are x=10 and y=4.


Submit your answer as: and

Solving simultaneous equations by elimination

Solve for x and y using the elimination method:

4x5y=1512x6y=18
Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]
Look at the coefficients of the x-terms. They are not equal, but one of them is a multiple of the other; that is the key to using the elimination method to solve simultaneous equations.
STEP: Start with the x-terms
[−2 points ⇒ 4 / 6 points left]

We need to solve the equations in this equation using elimination. (Substitution will also work, but the question says that we should use elimination.) As always for elimination, we need to use the two equations to cancel some of the terms.

By comparing the coefficients of the terms, we can see that the second x-term has a coefficient which is a multiple of the first x-term. In fact, 12 is 3 times as much as 4. We can make both coefficients of x the same by using this factor of 3 to multiply the entire first equation:

4x5y=15(3)4x(3)5y=15(3)12x15y=45

STEP: Eliminate the x-variable and solve for y
[−2 points ⇒ 2 / 6 points left]

Now we can eliminate the x-terms by subtracting one equation from the other. Then we can solve for y.

12x15y=45to cancel the x-terms:Subtract the equations(12x6y)=(18)0x9y=63

Now we can solve for y:

y=639=7

STEP: Substitute in for y
[−2 points ⇒ 0 / 6 points left]

The last step is to substitute y back into either of the equations so that we can find x. Here we will use the first equation:

4x5y=154x5(7)=154x+35=154x=20x=5

The answers are x=5 and y=7.


Submit your answer as: and

Solving simultaneous equations by elimination

Solve for x and y using the elimination method:

4x+y=73x+5y=18
Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]
Look at the coefficients of the y-terms. They are not equal, but one of them is a multiple of the other; that is the key to using the elimination method to solve simultaneous equations.
STEP: Start with the y-terms
[−2 points ⇒ 4 / 6 points left]

We need to solve the equations in this equation using elimination. (Substitution will also work, but the question says that we should use elimination.) As always for elimination, we need to use the two equations to cancel some of the terms.

By comparing the coefficients of the terms, we can see that the second y-term has a coefficient which is a multiple of the first y-term. In fact, 5 is 5 times as much as 1. We can make both coefficients of y the same by using this factor of 5 to multiply the entire first equation:

4x+y=7(5)4x+(5)y=7(5)20x+5y=35

STEP: Eliminate the y-variable and solve for x
[−2 points ⇒ 2 / 6 points left]

Now we can eliminate the y-terms by subtracting one equation from the other. Then we can solve for x.

20x+5y=35to cancel the y-terms:Subtract the equations(3x+5y)=(18)17x+0y=17

Now we can solve for x:

x=1717=1

STEP: Substitute in for x
[−2 points ⇒ 0 / 6 points left]

The last step is to substitute y back into either of the equations so that we can find x. Here we will use the first equation:

4x+y=74(1)+y=74+y=7y=3

The answers are x=1 and y=3.


Submit your answer as: and

Setting up elimination: picking a multiple

A friend in your maths class started to solve these two equations simultaneously:

4k+3p=73k5p=8

She is using the elimination method. She started by multiplying the second equation by 3, leading to this:

4k+3p=79k15p=24

But your friend is not sure what to do next, and she asks you for help. What number can you use to multiply the first equation so that the equations can be solved using elimination? Your answer should be an integer. Note that there may be more than one answer, but you should only give one answer.

Answer: You can multiply by .
one-of
type(numeric.noerror)
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Remember that the goal of elimination is to cancel either the k-terms or the p-terms. So you need to change one of the coefficients in the first equations so that you can cancel terms.


STEP: Find a way to make two of the terms cancel
[−1 point ⇒ 0 / 1 points left]

For this question we need to multiply the first equation by a number so that we can solve the equations by elimination. In the original equations, none of the coefficients were ready to cancel (because none of them are equal). The second equation was already multiplied by 3. We need to change one of the coefficients in the first equation so that we can cancel terms.

Specifically, we need to multiply 4 to get 9 so we can cancel the k-terms, or multiply the 3 to get 15 so we can cancel the p-terms.

The numbers in these equations point us toward cancelling the p-terms. This is because 15 is a multiple of 3. So if we multiply the first equation by 5 the the coefficients will be equal and opposite.

term in the equationmultiply each4(5)k+3(5)p=7(5)20k+15p=35

The coefficients we want to cancel are equal and opposite. So eliminating the terms requires adding both sides of the equation:

20k+15p=35equationadd this9k15p=2420k9k+15p15p=35+241k+0p=11
NOTE: If we had multiplied by 5, the signs in the first equation would be changed. Then we would subtract the equations instead. That means that 5 is also an acceptable answer.

The correct answer can be either of these numbers: 5 or 5.


Submit your answer as:

Setting up elimination: picking a multiple

A friend in your maths class started to solve these two equations simultaneously:

0=4a4b2=5a+3b

He is using the elimination method. He started by multiplying the second equation by 4, leading to this:

0=4a4b8=20a12b

But your friend is not sure what to do next, and he asks you for help. What number can you use to multiply the first equation so that the equations can be solved using elimination? Your answer should be an integer. Note that there may be more than one answer, but you should only give one answer.

Answer: You can multiply by .
one-of
type(numeric.noerror)
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Remember that the goal of elimination is to cancel either the a-terms or the b-terms. So you need to change one of the coefficients in the first equations so that you can cancel terms.


STEP: Find a way to make two of the terms cancel
[−1 point ⇒ 0 / 1 points left]

For this question we need to multiply the first equation by a number so that we can solve the equations by elimination. In the original equations, none of the coefficients were ready to cancel (because none of them are equal). The second equation was already multiplied by 4. We need to change one of the coefficients in the first equation so that we can cancel terms.

Specifically, we need to multiply 4 to get 20 so we can cancel the a-terms, or multiply the 4 to get 12 so we can cancel the b-terms.

For these equations, both options work. The coefficients of the a-terms and the b-terms in the second equation are multiples of the coefficients in the other equation. If we multiply by 3 we can cancel the b-terms. But we could also multiply by 5 which would allow us to cancel the a-terms. Both are good choices. Here we will multiply the first equation by 3.

term in the equationmultiply each0(3)=4(3)a4(3)b0=12a12b

The coefficients we want to cancel are equal. So eliminating the terms requires subtracting both sides of the equation:

0=12a12bequationsubtract this8=20a12b0+8=12a+20a12b+12b8=8a+0b
NOTE: If we had multiplied by 3, the signs in the first equation would be changed. Then we would add the equations instead. That means that 3 is also an acceptable answer.

The correct answer can be any of these numbers: 3, 3, 5 or 5.


Submit your answer as:

Setting up elimination: picking a multiple

A friend in your maths class started to solve these two equations simultaneously:

4k+2p=63k+3p=9

He is using the elimination method. He started by multiplying the second equation by 2, leading to this:

4k+2p=66k+6p=18

But your friend is not sure what to do next, and he asks you for help. What number can you use to multiply the first equation so that the equations can be solved using elimination? Your answer should be an integer. Note that there may be more than one answer, but you should only give one answer.

Answer: You can multiply by .
one-of
type(numeric.noerror)
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Remember that the goal of elimination is to cancel either the k-terms or the p-terms. So you need to change one of the coefficients in the first equations so that you can cancel terms.


STEP: Find a way to make two of the terms cancel
[−1 point ⇒ 0 / 1 points left]

For this question we need to multiply the first equation by a number so that we can solve the equations by elimination. In the original equations, none of the coefficients were ready to cancel (because none of them are equal). The second equation was already multiplied by 2. We need to change one of the coefficients in the first equation so that we can cancel terms.

Specifically, we need to multiply 4 to get 6 so we can cancel the k-terms, or multiply the 2 to get 6 so we can cancel the p-terms.

The numbers in these equations point us toward cancelling the p-terms. This is because 6 is a multiple of 2. So if we multiply the first equation by 3 the the coefficients will be equal.

term in the equationmultiply each4(3)k+2(3)p=6(3)12k+6p=18

The coefficients we want to cancel are equal. So eliminating the terms requires subtracting both sides of the equation:

12k+6p=18equationsubtract this6k+6p=1812k6k+6p6p=18+186k+0p=0
NOTE: If we had multiplied by 3, the signs in the first equation would be changed. Then we would add the equations instead. That means that 3 is also an acceptable answer.

The correct answer can be either of these numbers: 3 or 3.


Submit your answer as:

3. Substitution method

Solving simultaneous equations by substitution

Solve for x and y using substitution:

7x+4y=5and3x6y=3

Your answers should be exact (do not round off).

Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to solve these equations using the substitution method. So the first thing to do is pick one of the variables to isolate.


STEP: Prepare one of the equations for substitution
[−1 point ⇒ 3 / 4 points left]

The question tells us to solve the equations using the substitution method. That is not necessarily the best method, but we do not have a choice!

None of the variables are isolated, so we will need to isolate one of them for the substitution step. It would be convenient if there was a coefficient of 1 or 1, because that would make our work easier. But that is not the case. So we will just make x the subject of the first equation:

7x+4y=57x=4y5x=4y7+57

Unfortunately, this equation has 2 fractions in it. That is ugly, but it will work.


STEP: Substitute and solve for y
[−2 points ⇒ 1 / 4 points left]

Substitute the expression for x into the second equation. Then simplify and solve for the value of y:

3x6y=3in for xsubstitute3(4y7+57)6y=330y7+157=330y7=67y=15

STEP: Find the value of x
[−1 point ⇒ 0 / 4 points left]

Substitute this value of y back into one of the equations and solve for x. In this case, the second equation is better because it has fewer negatives than the first equation.

3x6y=33x6(15)=33x=95x=35

The final answers are x=35 and y=15.


Submit your answer as: and

Simultaneous equations: the substitution method

Here are two equations, which we can solve simultaneously using substitution:

4x=15+3yx=5y2

If we solve the equations using substitution, which variable is the best choice for the substitution, and why? Choose your answers from the choices below.

Answer: The best choice is to substitute because .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The best choice for substitution is based on which variable is easiest to isolate. In this case, you should look for the variable which is already isolated.


STEP: Look at the arrangement of the equations
[−2 points ⇒ 0 / 2 points left]

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is substitution. The substitution method works by using one of the equations to replace one of the variables in the other equation. We need to figure out which variable is the best choice to substitute for these two equations.

In these two equations, there are two x-terms and two y-terms:

4x=15+3yx=5y2

The goal of substitution is to use one of the equations to remove a variable from the other equation.

In this case, the best choice is to substitute the x from the second equation into the first equation. This is because in the second equation the x is already isolated. To use substitution, we need a variable which is isolated (alone on one side of the equation).

In this case, the substitution looks like this:

4x=15+3yx=5y2

Combining these equations by substitution, we get:

4(5y2)=15+3y

Since the second equation tells us that x is equal to x=5y2, we can substitute it into the other equation in place of x straight away.

The best choice is to substitute the x from the second equation into the first equation because it is already isolated.


Submit your answer as: and

Solving simultaneous equations by substitution

Solve simultaneously for x and y.

3x+2y=82x+y=6
TIP: Use the substitution method.
Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

The first thing you need to do (if you will use substitution to solve the pair of equations) is isolate one of the variables from one of the equations; then substitute what you get into the other equation.


STEP: Identify which variable to isolate
[−1 point ⇒ 5 / 6 points left]

When we solve simultaneous equations, we are working to find values for y and x which make both equations true!

We can solve these equations by substitution. Substitution requires isolating a variable. We need to look at the equations to decide which variable to isolate. We can see that the second equation looks good for this, because the y has a coefficient of one:

2x+y=6 isolationtarget for

STEP: Make y the subject of the equation
[−1 point ⇒ 4 / 6 points left]

Isolate y in the second equation, which can be done in one step:

2x+y=6y=2x+6

STEP: Substitute the result into the other equation to find x
[−2 points ⇒ 2 / 6 points left]

Now we can substitute the expression 2x+6 into the first equation and solve:

3x+2y=83x+2(2x+6)=83x4x+12=8x=4x=4

STEP: Use the x-value to find y
[−2 points ⇒ 0 / 6 points left]

Right now we have only half of the answer! To get the other half, we need to find the value of y. We can use the equation we got when we isolated y: it is the most convenient choice because y is the subject of the equation.

y=2x+6y=2(4)+6y=2

The correct answers are x=4 and y=2.


Submit your answer as: and

Exercises

Solving simultaneous equations by substitution

Solve for x and y using substitution:

8x+6y=8and9x4y=6

Your answers should be exact (do not round off).

Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to solve these equations using the substitution method. So the first thing to do is pick one of the variables to isolate.


STEP: Prepare one of the equations for substitution
[−1 point ⇒ 3 / 4 points left]

The question tells us to solve the equations using the substitution method. That is not necessarily the best method, but we do not have a choice!

None of the variables are isolated, so we will need to isolate one of them for the substitution step. It would be convenient if there was a coefficient of 1 or 1, because that would make our work easier. But that is not the case. So we will just make x the subject of the first equation:

8x+6y=88x=6y8x=3y41

Unfortunately, this equation has 1 fraction in it. That is ugly, but it will work.


STEP: Substitute and solve for y
[−2 points ⇒ 1 / 4 points left]

Substitute the expression for x into the second equation. Then simplify and solve for the value of y:

9x4y=6in for xsubstitute9(3y41)4y=611y4+9=611y4=15y=6011

STEP: Find the value of x
[−1 point ⇒ 0 / 4 points left]

Substitute this value of y back into one of the equations and solve for x. In this case, the first equation has only 1 negative sign while the other equation has 3 negatives. So we will use the first equation.

8x+6y=88x+6(6011)=88x=27211x=3411

The final answers are x=3411 and y=6011.


Submit your answer as: and

Solving simultaneous equations by substitution

Solve for x and y using substitution:

8x2y=2and4x4y=8

Your answers should be exact (do not round off).

Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to solve these equations using the substitution method. So the first thing to do is pick one of the variables to isolate.


STEP: Prepare one of the equations for substitution
[−1 point ⇒ 3 / 4 points left]

The question tells us to solve the equations using the substitution method. That is not necessarily the best method, but we do not have a choice!

None of the variables are isolated, so we will need to isolate one of them for the substitution step. It would be convenient if there was a coefficient of 1 or 1, because that would make our work easier. But that is not the case. So we will just make x the subject of the first equation:

8x2y=28x=2y2x=y4+14

Unfortunately, this equation has 2 fractions in it. That is ugly, but it will work.


STEP: Substitute and solve for y
[−2 points ⇒ 1 / 4 points left]

Substitute the expression for x into the second equation. Then simplify and solve for the value of y:

4x4y=8in for xsubstitute4(y4+14)4y=85y+1=85y=7y=75

STEP: Find the value of x
[−1 point ⇒ 0 / 4 points left]

Substitute this value of y back into one of the equations and solve for x. In this case, the second equation is better because it has fewer negatives than the first equation.

4x4y=84x4(75)=84x=125x=35

The final answers are x=35 and y=75.


Submit your answer as: and

Solving simultaneous equations by substitution

Solve for x and y using substitution:

2x4y=6and7x10y=6

Your answers should be exact (do not round off).

Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to solve these equations using the substitution method. So the first thing to do is pick one of the variables to isolate.


STEP: Prepare one of the equations for substitution
[−1 point ⇒ 3 / 4 points left]

The question tells us to solve the equations using the substitution method. That is not necessarily the best method, but we do not have a choice!

None of the variables are isolated, so we will need to isolate one of them for the substitution step. It would be convenient if there was a coefficient of 1 or 1, because that would make our work easier. But that is not the case. So we will just make x the subject of the first equation:

2x4y=62x=4y+6x=2y+3


STEP: Substitute and solve for y
[−2 points ⇒ 1 / 4 points left]

Substitute the expression for x into the second equation. Then simplify and solve for the value of y:

7x10y=6in for xsubstitute7(2y+3)10y=624y21=624y=27y=98

STEP: Find the value of x
[−1 point ⇒ 0 / 4 points left]

Substitute this value of y back into one of the equations and solve for x. In this case, the first equation has only 1 negative sign while the other equation has 2 negatives. So we will use the first equation.

2x4y=62x4(98)=62x=32x=34

The final answers are x=34 and y=98.


Submit your answer as: and

Simultaneous equations: the substitution method

Here are two equations, which we can solve simultaneously using substitution:

5y=5+5xy=2x+2

If we solve the equations using substitution, which variable is the best choice for the substitution, and why? Choose your answers from the choices below.

Answer: The best choice is to substitute because .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The best choice for substitution is based on which variable is easiest to isolate. In this case, you should look for the variable which is already isolated.


STEP: Look at the arrangement of the equations
[−2 points ⇒ 0 / 2 points left]

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is substitution. The substitution method works by using one of the equations to replace one of the variables in the other equation. We need to figure out which variable is the best choice to substitute for these two equations.

In these two equations, there are two x-terms and two y-terms:

5y=5+5xy=2x+2

The goal of substitution is to use one of the equations to remove a variable from the other equation.

In this case, the best choice is to substitute the y from the second equation into the first equation. This is because in the second equation the y is already isolated. To use substitution, we need a variable which is isolated (alone on one side of the equation).

In this case, the substitution looks like this:

5y=5+5xy=2x+2

Combining these equations by substitution, we get:

5(2x+2)=5+5x

Since the second equation tells us that y is equal to y=2x+2, we can substitute it into the other equation in place of y straight away.

The best choice is to substitute the y from the second equation into the first equation because it is already isolated.


Submit your answer as: and

Simultaneous equations: the substitution method

Here are two equations, which we can solve simultaneously using substitution:

y=3x45y=205x

If we solve the equations using substitution, which variable is the best choice for the substitution, and why? Choose your answers from the choices below.

Answer: The best choice is to substitute because .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The best choice for substitution is based on which variable is easiest to isolate. In this case, you should look for the variable which is already isolated.


STEP: Look at the arrangement of the equations
[−2 points ⇒ 0 / 2 points left]

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is substitution. The substitution method works by using one of the equations to replace one of the variables in the other equation. We need to figure out which variable is the best choice to substitute for these two equations.

In these two equations, there are two x-terms and two y-terms:

y=3x45y=205x

The goal of substitution is to use one of the equations to remove a variable from the other equation.

In this case, the best choice is to substitute the y from the first equation into the second equation. This is because in the first equation the y is already isolated. To use substitution, we need a variable which is isolated (alone on one side of the equation).

In this case, the substitution looks like this:

5y=205xy=3x4

Combining these equations by substitution, we get:

5(3x4)=205x

Since the first equation tells us that y is equal to y=3x4, we can substitute it into the other equation in place of y straight away.

The best choice is to substitute the y from the first equation into the second equation because it is already isolated.


Submit your answer as: and

Simultaneous equations: the substitution method

Here are two equations, which we can solve simultaneously using substitution:

x=5y+55x=33y

If we solve the equations using substitution, which variable is the best choice for the substitution, and why? Choose your answers from the choices below.

Answer: The best choice is to substitute because .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The best choice for substitution is based on which variable is easiest to isolate. In this case, you should look for the variable which is already isolated.


STEP: Look at the arrangement of the equations
[−2 points ⇒ 0 / 2 points left]

For this question, we are thinking about solving two equations 'simultaneously'. That means solving them together. One method for solving equations simultaneously is substitution. The substitution method works by using one of the equations to replace one of the variables in the other equation. We need to figure out which variable is the best choice to substitute for these two equations.

In these two equations, there are two x-terms and two y-terms:

x=5y+55x=33y

The goal of substitution is to use one of the equations to remove a variable from the other equation.

In this case, the best choice is to substitute the x from the first equation into the second equation. This is because in the first equation the x is already isolated. To use substitution, we need a variable which is isolated (alone on one side of the equation).

In this case, the substitution looks like this:

5x=33yx=5y+5

Combining these equations by substitution, we get:

5(5y+5)=33y

Since the first equation tells us that x is equal to x=5y+5, we can substitute it into the other equation in place of x straight away.

The best choice is to substitute the x from the first equation into the second equation because it is already isolated.


Submit your answer as: and

Solving simultaneous equations by substitution

Solve simultaneously for x and y.

5x+y=94x+3y=27
TIP: Use the substitution method.
Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

The first thing you need to do (if you will use substitution to solve the pair of equations) is isolate one of the variables from one of the equations; then substitute what you get into the other equation.


STEP: Identify which variable to isolate
[−1 point ⇒ 5 / 6 points left]

When we solve simultaneous equations, we are working to find values for y and x which make both equations true!

We can solve these equations by substitution. Substitution requires isolating a variable. We need to look at the equations to decide which variable to isolate. We can see that the first equation looks good for this, because the y has a coefficient of one:

5x+y=9 isolationtarget for

STEP: Make y the subject of the equation
[−1 point ⇒ 4 / 6 points left]

Isolate y in the first equation, which can be done in one step:

5x+y=9y=5x+9

STEP: Substitute the result into the other equation to find x
[−2 points ⇒ 2 / 6 points left]

Now we can substitute the expression 5x+9 into the second equation and solve:

4x+3y=274x+3(5x+9)=274x+15x+27=2719x=0x=0

STEP: Use the x-value to find y
[−2 points ⇒ 0 / 6 points left]

Right now we have only half of the answer! To get the other half, we need to find the value of y. We can use the equation we got when we isolated y: it is the most convenient choice because y is the subject of the equation.

y=5x+9y=5(0)+9y=9

The correct answers are x=0 and y=9.


Submit your answer as: and

Solving simultaneous equations by substitution

Solve simultaneously for x and y.

x+3y=173x+6y=36
TIP: Use the substitution method.
Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

The first thing you need to do (if you will use substitution to solve the pair of equations) is isolate one of the variables from one of the equations; then substitute what you get into the other equation.


STEP: Identify which variable to isolate
[−1 point ⇒ 5 / 6 points left]

When we solve simultaneous equations, we are working to find values for x and y which make both equations true!

We can solve these equations by substitution. Substitution requires isolating a variable. We need to look at the equations to decide which variable to isolate. We can see that the first equation looks good for this, because the x has a coefficient of one:

x+3y=17 isolationtarget for

STEP: Make x the subject of the equation
[−1 point ⇒ 4 / 6 points left]

Isolate x in the first equation, which can be done in one step:

x+3y=17x=3y+17

STEP: Substitute the result into the other equation to find y
[−2 points ⇒ 2 / 6 points left]

Now we can substitute the expression 3y+17 into the second equation and solve:

3x+6y=363(3y+17)+6y=369y+51+6y=363y=15y=5

STEP: Use the y-value to find x
[−2 points ⇒ 0 / 6 points left]

Right now we have only half of the answer! To get the other half, we need to find the value of x. We can use the equation we got when we isolated x: it is the most convenient choice because x is the subject of the equation.

x=3y+17x=3(5)+17x=2

The correct answers are x=2 and y=5.


Submit your answer as: and

Solving simultaneous equations by substitution

Solve simultaneously for x and y.

2x+6y=26x+y=32
TIP: Use the substitution method.
Answer: x= and y=
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

The first thing you need to do (if you will use substitution to solve the pair of equations) is isolate one of the variables from one of the equations; then substitute what you get into the other equation.


STEP: Identify which variable to isolate
[−1 point ⇒ 5 / 6 points left]

When we solve simultaneous equations, we are working to find values for y and x which make both equations true!

We can solve these equations by substitution. Substitution requires isolating a variable. We need to look at the equations to decide which variable to isolate. We can see that the second equation looks good for this, because the y has a coefficient of one:

6x+y=32 isolationtarget for

STEP: Make y the subject of the equation
[−1 point ⇒ 4 / 6 points left]

Isolate y in the second equation, which can be done in one step:

6x+y=32y=6x32

STEP: Substitute the result into the other equation to find x
[−2 points ⇒ 2 / 6 points left]

Now we can substitute the expression 6x32 into the first equation and solve:

2x+6y=22x+6(6x32)=22x36x192=238x=190x=5

STEP: Use the x-value to find y
[−2 points ⇒ 0 / 6 points left]

Right now we have only half of the answer! To get the other half, we need to find the value of y. We can use the equation we got when we isolated y: it is the most convenient choice because y is the subject of the equation.

y=6x32y=6(5)32y=2

The correct answers are x=5 and y=2.


Submit your answer as: and

4. Practical applications

Word problems: solving simultaneous equations

A group of friends is buying lunch together. The group buys 3 wraps and 4 sandwiches. Here are some facts about their lunch:

  • the total cost for the 3 wraps and 4 sandwiches is R178
  • a wrap costs R8 more than a sandwich

What is the price for one wrap and the price for one sandwich?

Answer:

A wrap costs R and a sandwich costs R .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 7 / 7 points left]

You need to choose variables to represent the things you want to find and write equations based on the information in the question.


STEP: Pick variables for the things we want to know
[−1 point ⇒ 6 / 7 points left]

The two things we want to know in this question are the prices for each of the items (a wrap and a sandwich). To begin, we can pick a variable for each of these numbers. It is helpful to pick variables which remind you about the things in the question:

w=the price of a wraps=the price of a sandwich

STEP: Write equations based on the information in the question
[−2 points ⇒ 4 / 7 points left]

Next we need to write equations based on what the question tells us. In other words, we need to translate the words in the question into equations.

The first point says that "the total cost for the 3 wraps and 4 sandwiches is R178." We can use the expression 3w to represent the price of the 3 wraps. Similarly, the expression 4s represents the price of the 4 sandwiches. With these values we can write a full equation for the prices:

In words: the total cost for the 3 wraps and 4 sandwiches is R178In maths: 3w+4s=178

Now we can use the second point. It says, "a wrap costs R8 more than a sandwich." We can write this as an equation like this:

In words: a wrap costs R8 more than a sandwichIn maths: w=s+8

STEP: Solve the equations simultaneously
[−2 points ⇒ 2 / 7 points left]

Now that we have two equations, we need to solve them simultaneously.

3w+4s=178w=s+8

We could use elimination, but substitution is a better choice (because w is already the subject). Substitute the second equation into the first equation and solve!

3w+4s=1783(s+8)+4s=1783s+24+4s=1787s=17824s=1547=22

This means that the price of one sandwich is R22.


STEP: Find the other variable's value
[−1 point ⇒ 1 / 7 points left]

Finally, use the value we found for s to find the value of w (the price of a wrap). We can use either equation to do this, but the second one is easier to use (because w is already isolated).

w=s+8=22+8=30

The price for one wrap is R30.


STEP: Write the final answer
[−1 point ⇒ 0 / 7 points left]

It is important to write the answer to a word problem as a complete sentence.

The price of the wrap is R30 while a sandwich costs R22.


Submit your answer as: and

Simultaneous equations

Solve the following equations simultaneously:

4x+y=144x+4y=16
Answer:

The numbers which solve both equations are x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Notice that the x-coefficients are opposites. You should start by adding the equations together, which will cancel those terms out.


STEP: Eliminate (cancel) the x-terms
[−1 point ⇒ 2 / 3 points left]

We have two equations and we need to solve them 'simultaneously'. That means we need to find two numbers, for x and for y, which solve both of the equations.

There are two methods for solving equations simultaneously: elimination and substitution. The equations in this question are perfect for elimination because the x-coefficients are equal and opposite. If we add the equations together, the x-terms will cancel. (We could solve these equations using the substitution method but elimination will be faster.)

So we add the equations together. This means we add together the left sides of the equations and also the right sides of the equations.

4x+y=144x+4y=164x4x+y+4y=14160x+5y=305y=30

STEP: Solve for y
[−1 point ⇒ 1 / 3 points left]

Now we have an equation with only one variable, and we can solve it.

5y=30y=6

STEP: Find the value of x
[−1 point ⇒ 0 / 3 points left]

So y=6. But we are not done yet: we also need an answer for x. We can get this using the answer we just got for y. Substitute this into either of the equations to get the answer - we can use either equation because we are looking for the number that solves both of them. We will pick the first equation, because it has fewer negative signs.

4x+y=144x+(6)=144x6=144x=8x=2

Now we have the complete answer: the numbers which solve the equations are x=2 and y=6. We can see this represented beautifully on a graph of the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are x=2 and y=6.


Submit your answer as: and

Word problems: test scores and simultaneous equations

John and Gina are friends. John takes Gina's accounting test paper and says: “I have 9 marks more than you do and the sum of both our marks is equal to 127. What are our marks?”

Answer:

John got marks and Gina got marks for the accounting test paper.

numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

Read through the question carefully and underline the important information. Then you will need to pick variables to represent each of the unknown facts from the question: the two marks. Use these variables to write down two equations which summarize the information in the question.


STEP: Pick variables for the marks
[−1 point ⇒ 5 / 6 points left]

We need to figure out the marks the students got on their tests. The first thing to do is to pick variables for each student's mark. It is helpful to pick variables which match the information we want, for example:

j= John's markg= Gina's mark

STEP: Write equations about the students' marks
[−2 points ⇒ 3 / 6 points left]

Now we can write equations with those variables. There are two pieces of information we have about the marks, which lead to two equations:

 more than GinaJohn has 9 marksj=g+9(the total) is 127The sum of the marksj+g=127

STEP: Solve the equations simultaneously
[−2 points ⇒ 1 / 6 points left]

We can solve these equations simultaneously. Substitute the first equation into the second equation and solve. (You can solve these equations using elimination if you prefer.)

j+g=127(g+9)+g=1272g=1279g=1182=59

This means that Gina's mark is 59.


STEP: Find John's mark
[−1 point ⇒ 0 / 6 points left]

Now we can use Gina's mark and one of the equations we have to find John's mark. The easiest way to do that is to substitute Gina's mark back into the first equation:

j=g+9=(59)+9=68

The students achieved these marks: John earned 68 and Gina earned 59.


Submit your answer as: and

Word problems: odd and even numbers

This question is about two positive numbers. Here are facts about these numbers:

  • The numbers are consecutive even integers.
  • The sum of the numbers is 38.

What is the value of the larger number?

Answer: The larger number is .
numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]
Start by choosing variables for the two numbers you are trying to find. For example, you can use n1 and n2. Then write equations using these variables based on the facts given in the question.
STEP: Pick variables
[−1 point ⇒ 4 / 5 points left]

This question is about two unknown numbers. And we need to find one of them (the larger one). We know certain things about these numbers: they are positive, they are consecutive even integers, and they have a sum of 38. We can solve this question using simultaneous equations.

The first thing to do is pick variables to represent the two numbers. Then we can write equations using those variables. It is usually helpful to pick variables which represent things we want to find. In this case, we are looking for two numbers, so these are good choices:

n1=the smaller numberwe need to findthis is the numbern2=the larger number

STEP: Write two equations
[−2 points ⇒ 2 / 5 points left]

The first fact given in the question says that the numbers are "consecutive even integers". So both of the numbers are even, and they come one after another. For example, 10 and 12 are consecutive even numbers. Since we defined n1 as the smaller number, n2 must be 2 more than n1.

n2=n1+2

The "+2" skips the odd integer which sits between n1 and n2.

The second fact about the numbers tells us that "the sum of the numbers is 38". Remember that sum means addition. So:

n1+n2=38

STEP: Solve the equations for n2
[−2 points ⇒ 0 / 5 points left]

We can now use these two equations to find the answer to the question. But remember that we only need to find the larger number (we do not need both of them). That means we need to find the value of n2.

We can solve this using substitution. However, remember that we want the value of n2 (the larger number). We can start by rearranging the first equation to make n1 the subject. Then the substitution step will remove n1 from the second equation and we can solve for n2. (This is not required - it just makes the solution faster.)

n2=n1+2n22=n1

Now substitute n22 into the other equation and solve for n2.

(n22)+n2=382n22=382n2=40n2=20

The result is n2=20. Notice that this means that the other number, n1, must be 18, because n2=n1+2. This is perfect, because we know that the sum of the numbers is 38, and 20+18=38.

The larger number is 20.


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Word problems: checking answers

Siyabonga is 14 years younger than his sister, Lesedi. In 9 years, Lesedi will be 2 times as old as Siyabonga. How old is Siyabonga now?

INSTRUCTION: If there is no acceptable solution, type 'no solution' in the answer box.
Answer:

Siyabonga is years old.

numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You need to find Siyabonga's age based on the given information. The first thing to do is define variables for the two ages, which are unknown. Then write equations with those variables based on the facts in the question.


STEP: Choose variables for each person's age
[−1 point ⇒ 4 / 5 points left]

We need to determine Siyabonga's age based on the information given in the question. To do this, we can write equations to represent the information, and then solve the equations.

The first thing to do is define variables for the unknowns in the question. The unknown values are the ages of both Lesedi and Siyabonga. We can use any variables we want, but it is best to choose variables which represent the information we want. In this case, we want ages, and we need to distinguish them somehow. Here is one good pair of options:

aL=Lesedi's ageaS=Siyabonga's age

STEP: Write an equation based on the information in the question
[−1 point ⇒ 3 / 5 points left]

Now let's write equations using these variables. From the question we know that: "Siyabonga is 14 years younger than his sister, Lesedi." We need to translate that into mathematics. The key word is younger, which tells us to use subtraction to relate the ages.

aS=aL14

STEP: Write the equations
[−1 point ⇒ 2 / 5 points left]

We also know that: "In 9 years, Lesedi will be 2 times as old as Siyabonga." Now we are looking into the future and we need to represent that information in maths. Using the variables we already have, we can write:

aL+9=Lesedi's age in 9 yearsaS+9=Siyabonga's age in 9 years

These are the ages at which Lesedi will be 2 times as old as Siyabonga. We can put all this together as follows:

in 9 yearsLesedi's age=2×in 9 yearsSiyabonga's ageaL+9=2(aS+9)

STEP: Solve the equations simultaneously
[−2 points ⇒ 0 / 5 points left]

Now we have to solve two equations, both of them including the variables aL and aS. The easiest way to solve them is substitution (you can use elimination if you prefer). The first equation is aS=aL14. We can substitute this into the second equation and solve for aL.

aL+9=2(aS+9)aL+9=2((aL14)+9)aL+9=2aL1019=aL

Terrific: this means that Lesedi is 19 years old. But the question asked for us to find Siyabonga's age. We can find it using the first equation, which relates the two ages:

aS=aL14=(19)14=5

So we finally got the answer to the question. Siyabonga is 5 years old.

The correct answer is: 5.


Submit your answer as:

Solving simultaneous equations

Solve the following equations simultaneously. Use whichever method is easiest.

x=4y+520=4x+3y
Answer:

The solution is x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You can solve the equations simultaneously using elimination or substitution. The first thing you should do is decide which method is the better choice for these equations.


STEP: Select the method for solving the question
[−1 point ⇒ 4 / 5 points left]

To solve the given equations simultaneously, we can use either substitution or elimination. The best choice depends on the arrangement of the equations. Since the x is isolated in the first equation, the best method here is substitution. We can substitute 4y+5 into the second equation in place of x.

NOTE: We can solve these equations using elimination - it will work. But it will probably be more complicated than substitution.

STEP: Do the substitution and solve for y
[−3 points ⇒ 1 / 5 points left]

Substitute the x and solve for y:

20=4x+3y20=4(4y+5)+3y20=13y2013y=0y=0

Super - we know that y=0.


STEP: Determine the other variable
[−1 point ⇒ 0 / 5 points left]

Now we can use the y-value to find the value of x. We can do this using either of the equations in the question. If one of the equations is simpler or more convenient (for example, if it has an isolated variable or has fewer negative signs) it is better to choose that equation. In this case, the first equation will be better, because x is already isolated.

x=4y+5x=4(0)+5x=5

The answer is the pair of numbers x=5 and y=0. On the Cartesian plane, this looks like this:

The values which solve these two equations are x=5 and y=0.


Submit your answer as: and

Word problems: an age-old question

Babatunde has a son, Mpho. Here are some facts about how old Babatunde and Mpho are:

  • Babatunde is 4 times as old as Mpho right now.
  • 7 years from now, Babatunde will be 3 times as old as Mpho.

How old are Babatunde and Mpho now?

Answer:

Babatunde is years old and Mpho is years old.

numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

Start by choosing variables for the things you want to know. Then write equations with those variables to summarise the information in the question.


STEP: Choose variables for the information
[−1 point ⇒ 5 / 6 points left]

In this question we want to find the age of two people: Babatunde and his son, Mpho. We can solve this by setting up two equations and solving them simultaneously.

Start by choosing variables to represent the ages of the father and the son. (We need to do this because we don't know the ages!) It is a good idea to choose variables that match what we are describing. So:

Let b=Babatunde's ageLet m=Mpho's age

STEP: Write equations based on the information in the question
[−2 points ⇒ 3 / 6 points left]

Now we want to use those variables to write equations. The first piece of information from the question tells us that "Babatunde is now 4 times as old as Mpho." As an equation, this is:

Ages now: b=4m

The second piece of information says that in "7 years... Babatunde will be 3 times as old as his son." In 7 years Babatunde will be b+7 years old, and similarly Mpho will be m+7 years old. Then:

Ages in 7 years: b+7=3(m+7)

STEP: Solve the equations simultaneously
[−2 points ⇒ 1 / 6 points left]

Now we have a pair of simultaneous equations! Substitute the first equation into the second equation and solve. (You can solve the equations using elimination if you prefer. In that case, the easiest option is to subtract the equations to cancel b.)

b+7=3(m+7)(4m)+7=3m+21m=14

Great! Now we know that Mpho is 14 years old.


STEP: Use Mpho's age to find Babatunde's age
[−1 point ⇒ 0 / 6 points left]

We can now find Babatunde's age. Substitute Mpho's age into one of the equations to do this. In this case, the first equation is the simpler choice for this calculation.

b=4m=4(14)=56

Write your final answer: Babatunde is 56 years old and Mpho is 14 years old.


Submit your answer as: and

Simultaneous equations

Solve the following equations simultaneously:

x=y+52x=12+3y
Answer: x= and y= .
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The first thing you need to do is substitute x=y+5 into the second equation.


STEP: Combine the equations using substitution
[−1 point ⇒ 3 / 4 points left]

We must solve the two equations 'simultaneously'. This means we want one pair of x and y values which solves both equations.

There are two methods for solving equations simultaneously: elimination and substitution. For the equations in this question, substitution is the better choice. This is because the first equation already shows us that x is equal to x=y+5, which is exactly what we need for the substitution method. (We could solve these equations using the elimination method, but it will require more work.)

So we can substitute y+5 into the second equation in place of x:

2x=12+3yx=y+52(y+5)=12+3y
NOTE: We could substitute the other variable if we want to; but the substitution above is the best choice because the x in the first equation is already the subject of the equation.

STEP: Solve for y
[−2 points ⇒ 1 / 4 points left]

Now that there is only one variable in the equation, we can solve the equation. Distribute the 2 and get on with solving the equation.

2(y+5)=12+3y2(y)2(5)=12+3y2y10=12+3y2y3y=12+10y=2y=2

STEP: Solve for x
[−1 point ⇒ 0 / 4 points left]

So y=2. But remember that we also need an answer for x. It is important to remember that you can use either equation to calculate x, because the numbers we want solve both equations. It is good to pick the easiest choice. In this case, the easiest choice is the first equation, because it is arranged in a more useful way.

x=y+5x=(2)+5x=2+5x=3

The numbers which solve the equations simultaneously are x=3 and y=2. We can see this represented beautifully if we graph the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are x=3 and y=2.


Submit your answer as: and

Picking a solution method

These two equations contain the same variables:

5x+2y=175x5y=5
  1. These equations can be solved using either the elimination or the substitution method. Which method is better for solving these two equations?

    Answer: The better method is .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    To decide which method is better, you should look at the coefficients. For example, can the coefficients lead to the cancellation of any of the terms in these equations?


    STEP: Check the arrangement of the equations to decide which method is better
    [−1 point ⇒ 0 / 1 points left]

    This question shows us two equations. We need to decide which method is the best choice for solving the equations simultaneously. (We do not have to solve the equations!)

    Two methods we can use for solving equations simultaneously are elimination and substitution. Both of those methods can be used to solve the equations in the question. But usually one of the methods is easier than the other based on the equations. Deciding which method is better depends on the arrangement of the equations:

    • Substitution is the better choice if there is already one variable isolated or if one variable is easy to isolate. If a variable is isolated, we can substitute immediately.
    • Elimination is the better choice if there are two terms which can be cancelled easily. If the terms can cancel, we can eliminate those terms immediately.

    The equations here are:

    5x+2y=175x5y=5

    In this case, the x-terms are ready to be eliminated. The coefficients of these terms are equal and opposite, which is perfect for the elimination method: it means we can cancel those two terms if we add the equations, which is how elimination works.

    The correct answer is: the elimination method.


    Submit your answer as:
  2. Solve the equations simultaneously.

    INSTRUCTION: Type your answer as a coordinate pair like this: (x;y).
    Answer: The answer is: .
    coordinate
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the method identified in Question 1.


    STEP: Eliminate (cancel) the x-terms
    [−1 point ⇒ 2 / 3 points left]

    Based on the result of Question 1, we should solve this question using elimination. It is possible to solve the equations using substitution, but using elimination is easier (as described in Question 1). Add the equations together to eliminate the x-terms. This means we add together the left and right sides of the equations.

    5x+2y=17equationadd this5x5y=55x5x+2y5y=17+50x3y=123y=12


    STEP: Solve for y
    [−1 point ⇒ 1 / 3 points left]

    Now we have an equation with only one variable, and we can solve it.

    3y=12y=4

    STEP: Find the value of x
    [−1 point ⇒ 0 / 3 points left]

    So y=4. But we also need an answer for x. We can get this using the answer we just got for y: substitute this into either of the equations to get the answer. We can use either equation because we are looking for the number that solves both of them. Since we have a choice, we should pick the easier equation (if there is one). We will pick the first equation, because it has fewer negative signs.

    5x+2y=175x+2(4)=175x+8=175x=25x=5

    Now we have the complete answer: the numbers which solve the equations simultaneously are x=5 and y=4.

    The correct answer is (5;4).


    Submit your answer as:

Setting up simultaneous equations

Last week, Abosede and Bongani had a chemistry test. Now they are comparing their marks and they notice these facts:

  • The sum of the marks is 135.
  • Abosede's mark is 15 less than Bongani's mark.

Let a represent Abosede's mark and b represent Bongani's mark. Then which equations below accurately represent the facts? Select your answer from the choices below.

Answer:
Fact about the test scores Equation
The sum of the marks is 135.
Abosede's mark is 15 less than Bongani's mark.
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

For the first equation, the key word is sum. For the second equation the key word is less. Use these key words to change the words into operations, for example, addition or multiplication.


STEP: Translate the first fact into an equation
[−1 point ⇒ 1 / 2 points left]

In this question we need to translate words into equations. This can be challenging. One useful approach is to look for important words which tell us what numbers and calculations to use. Here are some common words and what they mean when we write mathematical expressions and equations:

Word Meaning
sum +
product ×
is =
consecutive 1 apart
more than add to
less than subtract from

With these key words in mind, let's identify the key parts/words in each of these facts. Then we can translate each of the parts into maths.

The question tells us that we should use the variable a for Abosede's mark and b for Bongani's mark. So we can break up the first fact like this:

The sum of the marksis135a+b=135

The first fact is equivalent to this equation: a+b=135.


STEP: Translate the second fact into an equation
[−1 point ⇒ 0 / 2 points left]

Similarly, we can identify key parts of the second fact, and translate each into an expression.

Abosede's markis15 less than Bongani's marka=b15

This equation, a=b15, means that Abosede's mark is less that Bongani's mark.

The correct answers are:

Fact about the test scores Equation
The sum of the marks is 135. a+b=135
Abosede's mark is 15 less than Bongani's mark. a=b15

Submit your answer as: and

Exercises

Word problems: solving simultaneous equations

A group of friends is buying lunch together. The group buys 7 pizzas and 5 milkshakes. Here are some facts about their lunch:

  • the total cost for the 7 pizzas and 5 milkshakes is R342
  • a pizza costs R6 more than a milkshake

What is the price for one pizza and the price for one milkshake?

Answer:

A pizza costs R and a milkshake costs R .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 7 / 7 points left]

You need to choose variables to represent the things you want to find and write equations based on the information in the question.


STEP: Pick variables for the things we want to know
[−1 point ⇒ 6 / 7 points left]

The two things we want to know in this question are the prices for each of the items (a pizza and a milkshake). To begin, we can pick a variable for each of these numbers. It is helpful to pick variables which remind you about the things in the question:

p=the price of a pizzam=the price of a milkshake

STEP: Write equations based on the information in the question
[−2 points ⇒ 4 / 7 points left]

Next we need to write equations based on what the question tells us. In other words, we need to translate the words in the question into equations.

The first point says that "the total cost for the 7 pizzas and 5 milkshakes is R342." We can use the expression 7p to represent the price of the 7 pizzas. Similarly, the expression 5m represents the price of the 5 milkshakes. With these values we can write a full equation for the prices:

In words: the total cost for the 7 pizzas and 5 milkshakes is R342In maths: 7p+5m=342

Now we can use the second point. It says, "a pizza costs R6 more than a milkshake." We can write this as an equation like this:

In words: a pizza costs R6 more than a milkshakeIn maths: p=m+6

STEP: Solve the equations simultaneously
[−2 points ⇒ 2 / 7 points left]

Now that we have two equations, we need to solve them simultaneously.

7p+5m=342p=m+6

We could use elimination, but substitution is a better choice (because p is already the subject). Substitute the second equation into the first equation and solve!

7p+5m=3427(m+6)+5m=3427m+42+5m=34212m=34242m=30012=25

This means that the price of one milkshake is R25.


STEP: Find the other variable's value
[−1 point ⇒ 1 / 7 points left]

Finally, use the value we found for m to find the value of p (the price of a pizza). We can use either equation to do this, but the second one is easier to use (because p is already isolated).

p=m+6=25+6=31

The price for one pizza is R31.


STEP: Write the final answer
[−1 point ⇒ 0 / 7 points left]

It is important to write the answer to a word problem as a complete sentence.

The price of the pizza is R31 while a milkshake costs R25.


Submit your answer as: and

Word problems: solving simultaneous equations

A group of friends is buying lunch together. The group buys 5 wraps and 4 milkshakes. Here are some facts about their lunch:

  • the total cost for the 5 wraps and 4 milkshakes is R228
  • a wrap costs R6 more than a milkshake

What is the price for one wrap and the price for one milkshake?

Answer:

A wrap costs R and a milkshake costs R .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 7 / 7 points left]

You need to choose variables to represent the things you want to find and write equations based on the information in the question.


STEP: Pick variables for the things we want to know
[−1 point ⇒ 6 / 7 points left]

The two things we want to know in this question are the prices for each of the items (a wrap and a milkshake). To begin, we can pick a variable for each of these numbers. It is helpful to pick variables which remind you about the things in the question:

w=the price of a wrapm=the price of a milkshake

STEP: Write equations based on the information in the question
[−2 points ⇒ 4 / 7 points left]

Next we need to write equations based on what the question tells us. In other words, we need to translate the words in the question into equations.

The first point says that "the total cost for the 5 wraps and 4 milkshakes is R228." We can use the expression 5w to represent the price of the 5 wraps. Similarly, the expression 4m represents the price of the 4 milkshakes. With these values we can write a full equation for the prices:

In words: the total cost for the 5 wraps and 4 milkshakes is R228In maths: 5w+4m=228

Now we can use the second point. It says, "a wrap costs R6 more than a milkshake." We can write this as an equation like this:

In words: a wrap costs R6 more than a milkshakeIn maths: w=m+6

STEP: Solve the equations simultaneously
[−2 points ⇒ 2 / 7 points left]

Now that we have two equations, we need to solve them simultaneously.

5w+4m=228w=m+6

We could use elimination, but substitution is a better choice (because w is already the subject). Substitute the second equation into the first equation and solve!

5w+4m=2285(m+6)+4m=2285m+30+4m=2289m=22830m=1989=22

This means that the price of one milkshake is R22.


STEP: Find the other variable's value
[−1 point ⇒ 1 / 7 points left]

Finally, use the value we found for m to find the value of w (the price of a wrap). We can use either equation to do this, but the second one is easier to use (because w is already isolated).

w=m+6=22+6=28

The price for one wrap is R28.


STEP: Write the final answer
[−1 point ⇒ 0 / 7 points left]

It is important to write the answer to a word problem as a complete sentence.

The price of the wrap is R28 while a milkshake costs R22.


Submit your answer as: and

Word problems: solving simultaneous equations

A group of friends is buying lunch together. The group buys 4 salads and 6 hamburgers. Here are some facts about their lunch:

  • the total cost for the 4 salads and 6 hamburgers is R298
  • a salad costs R7 more than a hamburger

What is the price for one salad and the price for one hamburger?

Answer:

A salad costs R and a hamburger costs R .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 7 / 7 points left]

You need to choose variables to represent the things you want to find and write equations based on the information in the question.


STEP: Pick variables for the things we want to know
[−1 point ⇒ 6 / 7 points left]

The two things we want to know in this question are the prices for each of the items (a salad and a hamburger). To begin, we can pick a variable for each of these numbers. It is helpful to pick variables which remind you about the things in the question:

s=the price of a saladh=the price of a hamburger

STEP: Write equations based on the information in the question
[−2 points ⇒ 4 / 7 points left]

Next we need to write equations based on what the question tells us. In other words, we need to translate the words in the question into equations.

The first point says that "the total cost for the 4 salads and 6 hamburgers is R298." We can use the expression 4s to represent the price of the 4 salads. Similarly, the expression 6h represents the price of the 6 hamburgers. With these values we can write a full equation for the prices:

In words: the total cost for the 4 salads and 6 hamburgers is R298In maths: 4s+6h=298

Now we can use the second point. It says, "a salad costs R7 more than a hamburger." We can write this as an equation like this:

In words: a salad costs R7 more than a hamburgerIn maths: s=h+7

STEP: Solve the equations simultaneously
[−2 points ⇒ 2 / 7 points left]

Now that we have two equations, we need to solve them simultaneously.

4s+6h=298s=h+7

We could use elimination, but substitution is a better choice (because s is already the subject). Substitute the second equation into the first equation and solve!

4s+6h=2984(h+7)+6h=2984h+28+6h=29810h=29828h=27010=27

This means that the price of one hamburger is R27.


STEP: Find the other variable's value
[−1 point ⇒ 1 / 7 points left]

Finally, use the value we found for h to find the value of s (the price of a salad). We can use either equation to do this, but the second one is easier to use (because s is already isolated).

s=h+7=27+7=34

The price for one salad is R34.


STEP: Write the final answer
[−1 point ⇒ 0 / 7 points left]

It is important to write the answer to a word problem as a complete sentence.

The price of the salad is R34 while a hamburger costs R27.


Submit your answer as: and

Simultaneous equations

Solve the following equations simultaneously:

2=2x+y16=2x4y
Answer:

The numbers which solve both equations are x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Notice that the x-coefficients are opposites. You should start by adding the equations together, which will cancel those terms out.


STEP: Eliminate (cancel) the x-terms
[−1 point ⇒ 2 / 3 points left]

We have two equations and we need to solve them 'simultaneously'. That means we need to find two numbers, for x and for y, which solve both of the equations.

There are two methods for solving equations simultaneously: elimination and substitution. The equations in this question are perfect for elimination because the x-coefficients are equal and opposite. If we add the equations together, the x-terms will cancel. (We could solve these equations using the substitution method but elimination will be faster.)

So we add the equations together. This means we add together the left sides of the equations and also the right sides of the equations.

2=2x+y16=2x4y216=2x+2x+y4y18=0x3y18=3y

STEP: Solve for y
[−1 point ⇒ 1 / 3 points left]

Now we have an equation with only one variable, and we can solve it.

18=3y6=y

STEP: Find the value of x
[−1 point ⇒ 0 / 3 points left]

So y=6. But we are not done yet: we also need an answer for x. We can get this using the answer we just got for y. Substitute this into either of the equations to get the answer - we can use either equation because we are looking for the number that solves both of them. In this case, neither of the equations is much more friendly than the other, so we will just use the first equation.

2=2x+y2=2x+(6)2=2x+68=2x4=x

Now we have the complete answer: the numbers which solve the equations are x=4 and y=6. We can see this represented beautifully on a graph of the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are x=4 and y=6.


Submit your answer as: and

Simultaneous equations

Solve the following equations simultaneously:

4=4x+2y14=4x5y
Answer:

The numbers which solve both equations are x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Notice that the x-coefficients are opposites. You should start by adding the equations together, which will cancel those terms out.


STEP: Eliminate (cancel) the x-terms
[−1 point ⇒ 2 / 3 points left]

We have two equations and we need to solve them 'simultaneously'. That means we need to find two numbers, for x and for y, which solve both of the equations.

There are two methods for solving equations simultaneously: elimination and substitution. The equations in this question are perfect for elimination because the x-coefficients are equal and opposite. If we add the equations together, the x-terms will cancel. (We could solve these equations using the substitution method but elimination will be faster.)

So we add the equations together. This means we add together the left sides of the equations and also the right sides of the equations.

4=4x+2y14=4x5y414=4x+4x+2y5y18=0x3y18=3y

STEP: Solve for y
[−1 point ⇒ 1 / 3 points left]

Now we have an equation with only one variable, and we can solve it.

18=3y6=y

STEP: Find the value of x
[−1 point ⇒ 0 / 3 points left]

So y=6. But we are not done yet: we also need an answer for x. We can get this using the answer we just got for y. Substitute this into either of the equations to get the answer - we can use either equation because we are looking for the number that solves both of them. In this case, neither of the equations is much more friendly than the other, so we will just use the first equation.

4=4x+2y4=4x+2(6)4=4x+1216=4x4=x

Now we have the complete answer: the numbers which solve the equations are x=4 and y=6. We can see this represented beautifully on a graph of the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are x=4 and y=6.


Submit your answer as: and

Simultaneous equations

Solve the following equations simultaneously:

5x+5y=15x5y=3
Answer:

The numbers which solve both equations are x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Notice that the y-coefficients are opposites. You should start by adding the equations together, which will cancel those terms out.


STEP: Eliminate (cancel) the y-terms
[−1 point ⇒ 2 / 3 points left]

We have two equations and we need to solve them 'simultaneously'. That means we need to find two numbers, for x and for y, which solve both of the equations.

There are two methods for solving equations simultaneously: elimination and substitution. The equations in this question are perfect for elimination because the y-coefficients are equal and opposite. If we add the equations together, the y-terms will cancel. (We could solve these equations using the substitution method but elimination will be faster.)

So we add the equations together. This means we add together the left sides of the equations and also the right sides of the equations.

5x+5y=15x5y=35x+x+5y5y=15+36x+0y=126x=12

STEP: Solve for x
[−1 point ⇒ 1 / 3 points left]

Now we have an equation with only one variable, and we can solve it.

6x=12x=2

STEP: Find the value of y
[−1 point ⇒ 0 / 3 points left]

So x=2. But we are not done yet: we also need an answer for y. We can get this using the answer we just got for x. Substitute this into either of the equations to get the answer - we can use either equation because we are looking for the number that solves both of them. In this case, neither of the equations is much more friendly than the other, so we will just use the first equation.

5x+5y=155(2)+5y=1510+5y=155y=5y=1

Now we have the complete answer: the numbers which solve the equations are x=2 and y=1. We can see this represented beautifully on a graph of the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are x=2 and y=1.


Submit your answer as: and

Word problems: test scores and simultaneous equations

Abdulai and Melanie are friends. Abdulai takes Melanie's economics test paper and says: “I have 19 marks more than you do and the sum of both our marks is equal to 149. What are our marks?”

Answer:

Abdulai got marks and Melanie got marks for the economics test paper.

numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

Read through the question carefully and underline the important information. Then you will need to pick variables to represent each of the unknown facts from the question: the two marks. Use these variables to write down two equations which summarize the information in the question.


STEP: Pick variables for the marks
[−1 point ⇒ 5 / 6 points left]

We need to figure out the marks the students got on their tests. The first thing to do is to pick variables for each student's mark. It is helpful to pick variables which match the information we want, for example:

a= Abdulai's markm= Melanie's mark

STEP: Write equations about the students' marks
[−2 points ⇒ 3 / 6 points left]

Now we can write equations with those variables. There are two pieces of information we have about the marks, which lead to two equations:

 more than MelanieAbdulai has 19 marksa=m+19(the total) is 149The sum of the marksa+m=149

STEP: Solve the equations simultaneously
[−2 points ⇒ 1 / 6 points left]

We can solve these equations simultaneously. Substitute the first equation into the second equation and solve. (You can solve these equations using elimination if you prefer.)

a+m=149(m+19)+m=1492m=14919m=1302=65

This means that Melanie's mark is 65.


STEP: Find Abdulai's mark
[−1 point ⇒ 0 / 6 points left]

Now we can use Melanie's mark and one of the equations we have to find Abdulai's mark. The easiest way to do that is to substitute Melanie's mark back into the first equation:

a=m+19=(65)+19=84

The students achieved these marks: Abdulai earned 84 and Melanie earned 65.


Submit your answer as: and

Word problems: test scores and simultaneous equations

Realeboha and Mathe are friends. Realeboha takes Mathe's life sciences test paper and says: “I have 20 marks more than you do and the sum of both our marks is equal to 126. What are our marks?”

Answer:

Realeboha got marks and Mathe got marks for the life sciences test paper.

numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

Read through the question carefully and underline the important information. Then you will need to pick variables to represent each of the unknown facts from the question: the two marks. Use these variables to write down two equations which summarize the information in the question.


STEP: Pick variables for the marks
[−1 point ⇒ 5 / 6 points left]

We need to figure out the marks the students got on their tests. The first thing to do is to pick variables for each student's mark. It is helpful to pick variables which match the information we want, for example:

r= Realeboha's markm= Mathe's mark

STEP: Write equations about the students' marks
[−2 points ⇒ 3 / 6 points left]

Now we can write equations with those variables. There are two pieces of information we have about the marks, which lead to two equations:

 more than MatheRealeboha has 20 marksr=m+20(the total) is 126The sum of the marksr+m=126

STEP: Solve the equations simultaneously
[−2 points ⇒ 1 / 6 points left]

We can solve these equations simultaneously. Substitute the first equation into the second equation and solve. (You can solve these equations using elimination if you prefer.)

r+m=126(m+20)+m=1262m=12620m=1062=53

This means that Mathe's mark is 53.


STEP: Find Realeboha's mark
[−1 point ⇒ 0 / 6 points left]

Now we can use Mathe's mark and one of the equations we have to find Realeboha's mark. The easiest way to do that is to substitute Mathe's mark back into the first equation:

r=m+20=(53)+20=73

The students achieved these marks: Realeboha earned 73 and Mathe earned 53.


Submit your answer as: and

Word problems: test scores and simultaneous equations

Yengwayo and Johan are friends. Yengwayo takes Johan's physics test paper and says: “I have 14 marks more than you do and the sum of both our marks is equal to 156. What are our marks?”

Answer:

Yengwayo got marks and Johan got marks for the physics test paper.

numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

Read through the question carefully and underline the important information. Then you will need to pick variables to represent each of the unknown facts from the question: the two marks. Use these variables to write down two equations which summarize the information in the question.


STEP: Pick variables for the marks
[−1 point ⇒ 5 / 6 points left]

We need to figure out the marks the students got on their tests. The first thing to do is to pick variables for each student's mark. It is helpful to pick variables which match the information we want, for example:

y= Yengwayo's markj= Johan's mark

STEP: Write equations about the students' marks
[−2 points ⇒ 3 / 6 points left]

Now we can write equations with those variables. There are two pieces of information we have about the marks, which lead to two equations:

 more than JohanYengwayo has 14 marksy=j+14(the total) is 156The sum of the marksy+j=156

STEP: Solve the equations simultaneously
[−2 points ⇒ 1 / 6 points left]

We can solve these equations simultaneously. Substitute the first equation into the second equation and solve. (You can solve these equations using elimination if you prefer.)

y+j=156(j+14)+j=1562j=15614j=1422=71

This means that Johan's mark is 71.


STEP: Find Yengwayo's mark
[−1 point ⇒ 0 / 6 points left]

Now we can use Johan's mark and one of the equations we have to find Yengwayo's mark. The easiest way to do that is to substitute Johan's mark back into the first equation:

y=j+14=(71)+14=85

The students achieved these marks: Yengwayo earned 85 and Johan earned 71.


Submit your answer as: and

Word problems: odd and even numbers

This question is about two positive numbers. Here are facts about these numbers:

  • The numbers are consecutive even integers.
  • The sum of the numbers is 38.

What is the value of the smaller number?

Answer: The smaller number is .
numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]
Start by choosing variables for the two numbers you are trying to find. For example, you can use n1 and n2. Then write equations using these variables based on the facts given in the question.
STEP: Pick variables
[−1 point ⇒ 4 / 5 points left]

This question is about two unknown numbers. And we need to find one of them (the smaller one). We know certain things about these numbers: they are positive, they are consecutive even integers, and they have a sum of 38. We can solve this question using simultaneous equations.

The first thing to do is pick variables to represent the two numbers. Then we can write equations using those variables. It is usually helpful to pick variables which represent things we want to find. In this case, we are looking for two numbers, so these are good choices:

we need to findthis is the numbern1=the smaller numbern2=the larger number

STEP: Write two equations
[−2 points ⇒ 2 / 5 points left]

The first fact given in the question says that the numbers are "consecutive even integers". So both of the numbers are even, and they come one after another. For example, 10 and 12 are consecutive even numbers. Since we defined n1 as the smaller number, n2 must be 2 more than n1.

n2=n1+2

The "+2" skips the odd integer which sits between n1 and n2.

The second fact about the numbers tells us that "the sum of the numbers is 38". Remember that sum means addition. So:

n1+n2=38

STEP: Solve the equations for n1
[−2 points ⇒ 0 / 5 points left]

We can now use these two equations to find the answer to the question. But remember that we only need to find the smaller number (we do not need both of them). That means we need to find the value of n1.

We can do this using substitution. If we substitute n2=n1+2 into the equation n1+n2=38, the n2 terms will disappear, leaving n1. Then we can solve for n1, which is exactly what we want.

n1+(n1+2)=382n1+2=382n1=36n1=18

The result is n1=18. Notice that this means that the other number, n2, must be 20, because n2=n1+2. This is perfect, because we know that the sum of the numbers is 38, and 18+20=38.

The smaller number is 18.


Submit your answer as:

Word problems: odd and even numbers

This question is about two positive numbers. Here are facts about these numbers:

  • The numbers are consecutive even integers.
  • The sum of the numbers is 34.

What is the value of the larger number?

Answer: The larger number is .
numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]
Start by choosing variables for the two numbers you are trying to find. For example, you can use n1 and n2. Then write equations using these variables based on the facts given in the question.
STEP: Pick variables
[−1 point ⇒ 4 / 5 points left]

This question is about two unknown numbers. And we need to find one of them (the larger one). We know certain things about these numbers: they are positive, they are consecutive even integers, and they have a sum of 34. We can solve this question using simultaneous equations.

The first thing to do is pick variables to represent the two numbers. Then we can write equations using those variables. It is usually helpful to pick variables which represent things we want to find. In this case, we are looking for two numbers, so these are good choices:

n1=the smaller numberwe need to findthis is the numbern2=the larger number

STEP: Write two equations
[−2 points ⇒ 2 / 5 points left]

The first fact given in the question says that the numbers are "consecutive even integers". So both of the numbers are even, and they come one after another. For example, 10 and 12 are consecutive even numbers. Since we defined n1 as the smaller number, n2 must be 2 more than n1.

n2=n1+2

The "+2" skips the odd integer which sits between n1 and n2.

The second fact about the numbers tells us that "the sum of the numbers is 34". Remember that sum means addition. So:

n1+n2=34

STEP: Solve the equations for n2
[−2 points ⇒ 0 / 5 points left]

We can now use these two equations to find the answer to the question. But remember that we only need to find the larger number (we do not need both of them). That means we need to find the value of n2.

We can solve this using substitution. However, remember that we want the value of n2 (the larger number). We can start by rearranging the first equation to make n1 the subject. Then the substitution step will remove n1 from the second equation and we can solve for n2. (This is not required - it just makes the solution faster.)

n2=n1+2n22=n1

Now substitute n22 into the other equation and solve for n2.

(n22)+n2=342n22=342n2=36n2=18

The result is n2=18. Notice that this means that the other number, n1, must be 16, because n2=n1+2. This is perfect, because we know that the sum of the numbers is 34, and 18+16=34.

The larger number is 18.


Submit your answer as:

Word problems: odd and even numbers

This question is about two positive numbers. Here are facts about these numbers:

  • The numbers are consecutive odd integers.
  • The sum of the numbers is 32.

What is the value of the smaller number?

Answer: The smaller number is .
numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]
Start by choosing variables for the two numbers you are trying to find. For example, you can use n1 and n2. Then write equations using these variables based on the facts given in the question.
STEP: Pick variables
[−1 point ⇒ 4 / 5 points left]

This question is about two unknown numbers. And we need to find one of them (the smaller one). We know certain things about these numbers: they are positive, they are consecutive odd integers, and they have a sum of 32. We can solve this question using simultaneous equations.

The first thing to do is pick variables to represent the two numbers. Then we can write equations using those variables. It is usually helpful to pick variables which represent things we want to find. In this case, we are looking for two numbers, so these are good choices:

we need to findthis is the numbern1=the smaller numbern2=the larger number

STEP: Write two equations
[−2 points ⇒ 2 / 5 points left]

The first fact given in the question says that the numbers are "consecutive odd integers". So both of the numbers are odd, and they come one after another. For example, 11 and 13 are consecutive odd numbers. Since we defined n1 as the smaller number, n2 must be 2 more than n1.

n2=n1+2

The "+2" skips the even integer which sits between n1 and n2.

The second fact about the numbers tells us that "the sum of the numbers is 32". Remember that sum means addition. So:

n1+n2=32

STEP: Solve the equations for n1
[−2 points ⇒ 0 / 5 points left]

We can now use these two equations to find the answer to the question. But remember that we only need to find the smaller number (we do not need both of them). That means we need to find the value of n1.

We can do this using substitution. If we substitute n2=n1+2 into the equation n1+n2=32, the n2 terms will disappear, leaving n1. Then we can solve for n1, which is exactly what we want.

n1+(n1+2)=322n1+2=322n1=30n1=15

The result is n1=15. Notice that this means that the other number, n2, must be 17, because n2=n1+2. This is perfect, because we know that the sum of the numbers is 32, and 15+17=32.

The smaller number is 15.


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Word problems: checking answers

Nkiru is 12 years younger than her sister, Adaeze. In 10 years, Adaeze will be 7 times as old as Nkiru. How old is Nkiru now?

INSTRUCTION: If there is no acceptable solution, type 'no solution' in the answer box.
Answer:

Nkiru is years old.

one-of
type(string.nocase)
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You need to find Nkiru's age based on the given information. The first thing to do is define variables for the two ages, which are unknown. Then write equations with those variables based on the facts in the question.


STEP: Choose variables for each person's age
[−1 point ⇒ 4 / 5 points left]

We need to determine Nkiru's age based on the information given in the question. To do this, we can write equations to represent the information, and then solve the equations.

The first thing to do is define variables for the unknowns in the question. The unknown values are the ages of both Adaeze and Nkiru. We can use any variables we want, but it is best to choose variables which represent the information we want. In this case, we want ages, and we need to distinguish them somehow. Here is one good pair of options:

aA=Adaeze's ageaN=Nkiru's age

STEP: Write an equation based on the information in the question
[−1 point ⇒ 3 / 5 points left]

Now let's write equations using these variables. From the question we know that: "Nkiru is 12 years younger than her sister, Adaeze." We need to translate that into mathematics. The key word is younger, which tells us to use subtraction to relate the ages.

aN=aA12

STEP: Write the equations
[−1 point ⇒ 2 / 5 points left]

We also know that: "In 10 years, Adaeze will be 7 times as old as Nkiru." Now we are looking into the future and we need to represent that information in maths. Using the variables we already have, we can write:

aA+10=Adaeze's age in 10 yearsaN+10=Nkiru's age in 10 years

These are the ages at which Adaeze will be 7 times as old as Nkiru. We can put all this together as follows:

in 10 yearsAdaeze's age=7×in 10 yearsNkiru's ageaA+10=7(aN+10)

STEP: Solve the equations simultaneously
[−2 points ⇒ 0 / 5 points left]

Now we have to solve two equations, both of them including the variables aA and aN. The easiest way to solve them is substitution (you can use elimination if you prefer). The first equation is aN=aA12. We can substitute this into the second equation and solve for aA.

aA+10=7(aN+10)aA+10=7((aA12)+10)aA+10=7aA1424=6aA4=aA

Terrific: this means that Adaeze is 4 years old. But the question asked for us to find Nkiru's age. We can find it using the first equation, which relates the two ages:

aN=aA12=(4)12=8

So we finally got the answer to the question. But wait a minute: aN represents the age of a person. It can't be negative! This either means that we made a mistake, or that there is no solution to the question. There is no mistake in the work: it turns out that the facts given about the two people's ages are not possible! The numbers agree with all the relationships given in the question, but we cannot forget that the numbers in this question have meaning. They refer to how many years it has been since someone was born. And that cannot be a negative number.

The correct answer is: no solution.


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Word problems: checking answers

Adedapo is 18 years older than his sister, Talatu. In 8 years, Adedapo will be 3 times as old as Talatu. How old is Talatu now?

INSTRUCTION: If there is no acceptable solution, type 'no solution' in the answer box.
Answer:

Talatu is years old.

numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You need to find Talatu's age based on the given information. The first thing to do is define variables for the two ages, which are unknown. Then write equations with those variables based on the facts in the question.


STEP: Choose variables for each person's age
[−1 point ⇒ 4 / 5 points left]

We need to determine Talatu's age based on the information given in the question. To do this, we can write equations to represent the information, and then solve the equations.

The first thing to do is define variables for the unknowns in the question. The unknown values are the ages of both Adedapo and Talatu. We can use any variables we want, but it is best to choose variables which represent the information we want. In this case, we want ages, and we need to distinguish them somehow. Here is one good pair of options:

aA=Adedapo's ageaT=Talatu's age

STEP: Write an equation based on the information in the question
[−1 point ⇒ 3 / 5 points left]

Now let's write equations using these variables. From the question we know that: "Adedapo is 18 years older than his sister, Talatu." We need to translate that into mathematics. The key word is older, which tells us to use addition to relate the ages.

aA=aT+18

STEP: Write the equations
[−1 point ⇒ 2 / 5 points left]

We also know that: "In 8 years, Adedapo will be 3 times as old as Talatu." Now we are looking into the future and we need to represent that information in maths. Using the variables we already have, we can write:

aA+8=Adedapo's age in 8 yearsaT+8=Talatu's age in 8 years

These are the ages at which Adedapo will be 3 times as old as Talatu. We can put all this together as follows:

in 8 yearsAdedapo's age=3×in 8 yearsTalatu's ageaA+8=3(aT+8)

STEP: Solve the equations simultaneously
[−2 points ⇒ 0 / 5 points left]

Now we have to solve two equations, both of them including the variables aA and aT. The easiest way to solve them is substitution (you can use elimination if you prefer). The first equation is aA=aT+18. We can substitute this into the second equation and solve for aT.

aA+8=3(aT+8)(aT+18)+8=3(aT+8)aT+26=3aT+242=2aT1=aT

Terrific: we have the answer.

Talatu is 1 year old.

The correct answer is: 1.


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Word problems: checking answers

Kelly is 9 years older than her sister, Justine. In 5 years, Kelly will be 2 times as old as Justine. How old is Justine now?

INSTRUCTION: If there is no acceptable solution, type 'no solution' in the answer box.
Answer:

Justine is years old.

numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You need to find Justine's age based on the given information. The first thing to do is define variables for the two ages, which are unknown. Then write equations with those variables based on the facts in the question.


STEP: Choose variables for each person's age
[−1 point ⇒ 4 / 5 points left]

We need to determine Justine's age based on the information given in the question. To do this, we can write equations to represent the information, and then solve the equations.

The first thing to do is define variables for the unknowns in the question. The unknown values are the ages of both Kelly and Justine. We can use any variables we want, but it is best to choose variables which represent the information we want. In this case, we want ages, and we need to distinguish them somehow. Here is one good pair of options:

aK=Kelly's ageaJ=Justine's age

STEP: Write an equation based on the information in the question
[−1 point ⇒ 3 / 5 points left]

Now let's write equations using these variables. From the question we know that: "Kelly is 9 years older than her sister, Justine." We need to translate that into mathematics. The key word is older, which tells us to use addition to relate the ages.

aK=aJ+9

STEP: Write the equations
[−1 point ⇒ 2 / 5 points left]

We also know that: "In 5 years, Kelly will be 2 times as old as Justine." Now we are looking into the future and we need to represent that information in maths. Using the variables we already have, we can write:

aK+5=Kelly's age in 5 yearsaJ+5=Justine's age in 5 years

These are the ages at which Kelly will be 2 times as old as Justine. We can put all this together as follows:

in 5 yearsKelly's age=2×in 5 yearsJustine's ageaK+5=2(aJ+5)

STEP: Solve the equations simultaneously
[−2 points ⇒ 0 / 5 points left]

Now we have to solve two equations, both of them including the variables aK and aJ. The easiest way to solve them is substitution (you can use elimination if you prefer). The first equation is aK=aJ+9. We can substitute this into the second equation and solve for aJ.

aK+5=2(aJ+5)(aJ+9)+5=2(aJ+5)aJ+14=2aJ+104=aJ

Terrific: we have the answer.

Justine is 4 years old.

The correct answer is: 4.


Submit your answer as:

Solving simultaneous equations

Solve the following equations simultaneously. Use whichever method is easiest.

2y16=3x4y+12=x
Answer:

The solution is x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You can solve the equations simultaneously using elimination or substitution. The first thing you should do is decide which method is the better choice for these equations.


STEP: Select the method for solving the question
[−1 point ⇒ 4 / 5 points left]

To solve the given equations simultaneously, we can use either substitution or elimination. The best choice depends on the arrangement of the equations. Since the x is isolated in the second equation, the best method here is substitution. We can substitute 4y+12 into the first equation in place of x.

NOTE: We can solve these equations using elimination - it will work. But it will probably be more complicated than substitution.

STEP: Do the substitution and solve for y
[−3 points ⇒ 1 / 5 points left]

Substitute the x and solve for y:

2y16=3x2y16=3(4y+12)2y16=12y3610y=20y=2

Super - we know that y=2.


STEP: Determine the other variable
[−1 point ⇒ 0 / 5 points left]

Now we can use the y-value to find the value of x. We can do this using either of the equations in the question. If one of the equations is simpler or more convenient (for example, if it has an isolated variable or has fewer negative signs) it is better to choose that equation. In this case, the second equation will be better, because x is already isolated.

4y+12=x4(2)+12=xx=4x=4

The answer is the pair of numbers x=4 and y=2. On the Cartesian plane, this looks like this:

The values which solve these two equations are x=4 and y=2.


Submit your answer as: and

Solving simultaneous equations

Solve the following equations simultaneously. Use whichever method is easiest.

3x=y+204x+3y=8
Answer:

The solution is x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

You can solve the equations simultaneously using elimination or substitution. The first thing you should do is decide which method is the better choice for these equations.


STEP: Select the method for solving the question
[−1 point ⇒ 5 / 6 points left]

To solve the given equations simultaneously, we can use either substitution or elimination. The best choice depends on the arrangement of the equations. In this case we need to modify the equations before we can do elimination or substitution. And we only need to make a small change to arrange the equations for substitution: we just need to subtract the 20 term to the other side of the first equation.

NOTE: We can solve these equations using elimination - it will work. But it will probably be more complicated than substitution.

STEP: Do the substitution and solve for x
[−4 points ⇒ 1 / 6 points left]

First rearrange the first equation to set up the substitution:

3x=y+203x20=y

Now do the substitution and complete the solution to find x.

4x+3y=84x+3(3x20)=813x60=813x=52x=4

Super - we know that x=4.


STEP: Determine the other variable
[−1 point ⇒ 0 / 6 points left]

Now we can use the x-value to find the value of y. We can do this using either of the equations in the question. If one of the equations is simpler or more convenient (for example, if it has an isolated variable or has fewer negative signs) it is better to choose that equation. In this case, the second equation is better because it has fewer negative signs.

4x+3y=84(4)+3y=83y=24y=8

The answer is the pair of numbers x=4 and y=8. On the Cartesian plane, this looks like this:

The values which solve these two equations are x=4 and y=8.


Submit your answer as: and

Solving simultaneous equations

Solve the following equations simultaneously. Use whichever method is easiest.

5x9=4y3x=3y+6
Answer:

The solution is x= and y= .

numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

You can solve the equations simultaneously using elimination or substitution. The first thing you should do is decide which method is the better choice for these equations.


STEP: Select the method for solving the question
[−1 point ⇒ 5 / 6 points left]

To solve the given equations simultaneously, we can use either substitution or elimination. The best choice depends on the arrangement of the equations. These equations are not arranged nicely for elimination or substitution. However, notice that the coefficients in the second equation are all a multiple of 3. So we can divide the equation by 3 to isolate x without picking up any fractions!

NOTE: We can solve these equations using elimination - it will work. But it will probably be more complicated than substitution.

STEP: Do the substitution and solve for y
[−4 points ⇒ 1 / 6 points left]

First rearrange the second equation to set up the substitution:

3x=3y+6x=y2

Now do the substitution and complete the solution to find y.

5x9=4y5(y2)9=4y5y+1=4yy=1y=1

Super - we know that y=1.


STEP: Determine the other variable
[−1 point ⇒ 0 / 6 points left]

Now we can use the y-value to find the value of x. We can do this using either of the equations in the question. If one of the equations is simpler or more convenient (for example, if it has an isolated variable or has fewer negative signs) it is better to choose that equation. In this case, the second equation is better because it has fewer negative signs.

3x=3y+63x=3(1)+63x=3x=1

The answer is the pair of numbers x=1 and y=1. On the Cartesian plane, this looks like this:

The values which solve these two equations are x=1 and y=1.


Submit your answer as: and

Word problems: an age-old question

Dumile has a son, Anathi. Here are some facts about how old Dumile and Anathi are:

  • Dumile is 11 times as old as Anathi right now.
  • 5 years from now, Dumile will be 6 times as old as Anathi.

How old are Dumile and Anathi now?

Answer:

Dumile is years old and Anathi is years old.

numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

Start by choosing variables for the things you want to know. Then write equations with those variables to summarise the information in the question.


STEP: Choose variables for the information
[−1 point ⇒ 5 / 6 points left]

In this question we want to find the age of two people: Dumile and his son, Anathi. We can solve this by setting up two equations and solving them simultaneously.

Start by choosing variables to represent the ages of the father and the son. (We need to do this because we don't know the ages!) It is a good idea to choose variables that match what we are describing. So:

Let d=Dumile's ageLet a=Anathi's age

STEP: Write equations based on the information in the question
[−2 points ⇒ 3 / 6 points left]

Now we want to use those variables to write equations. The first piece of information from the question tells us that "Dumile is now 11 times as old as Anathi." As an equation, this is:

Ages now: d=11a

The second piece of information says that in "5 years... Dumile will be 6 times as old as his son." In 5 years Dumile will be d+5 years old, and similarly Anathi will be a+5 years old. Then:

Ages in 5 years: d+5=6(a+5)

STEP: Solve the equations simultaneously
[−2 points ⇒ 1 / 6 points left]

Now we have a pair of simultaneous equations! Substitute the first equation into the second equation and solve. (You can solve the equations using elimination if you prefer. In that case, the easiest option is to subtract the equations to cancel d.)

d+5=6(a+5)(11a)+5=6a+305a=25a=5

Great! Now we know that Anathi is 5 years old.


STEP: Use Anathi's age to find Dumile's age
[−1 point ⇒ 0 / 6 points left]

We can now find Dumile's age. Substitute Anathi's age into one of the equations to do this. In this case, the first equation is the simpler choice for this calculation.

d=11a=11(5)=55

Write your final answer: Dumile is 55 years old and Anathi is 5 years old.


Submit your answer as: and

Word problems: an age-old question

Phetoho has a son, Azubuike. Here are some facts about how old Phetoho and Azubuike are:

  • Phetoho is 6 times as old as Azubuike right now.
  • 9 years from now, Phetoho will be 3 times as old as Azubuike.

How old are Phetoho and Azubuike now?

Answer:

Phetoho is years old and Azubuike is years old.

numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

Start by choosing variables for the things you want to know. Then write equations with those variables to summarise the information in the question.


STEP: Choose variables for the information
[−1 point ⇒ 5 / 6 points left]

In this question we want to find the age of two people: Phetoho and his son, Azubuike. We can solve this by setting up two equations and solving them simultaneously.

Start by choosing variables to represent the ages of the father and the son. (We need to do this because we don't know the ages!) It is a good idea to choose variables that match what we are describing. So:

Let p=Phetoho's ageLet a=Azubuike's age

STEP: Write equations based on the information in the question
[−2 points ⇒ 3 / 6 points left]

Now we want to use those variables to write equations. The first piece of information from the question tells us that "Phetoho is now 6 times as old as Azubuike." As an equation, this is:

Ages now: p=6a

The second piece of information says that in "9 years... Phetoho will be 3 times as old as his son." In 9 years Phetoho will be p+9 years old, and similarly Azubuike will be a+9 years old. Then:

Ages in 9 years: p+9=3(a+9)

STEP: Solve the equations simultaneously
[−2 points ⇒ 1 / 6 points left]

Now we have a pair of simultaneous equations! Substitute the first equation into the second equation and solve. (You can solve the equations using elimination if you prefer. In that case, the easiest option is to subtract the equations to cancel p.)

p+9=3(a+9)(6a)+9=3a+273a=18a=6

Great! Now we know that Azubuike is 6 years old.


STEP: Use Azubuike's age to find Phetoho's age
[−1 point ⇒ 0 / 6 points left]

We can now find Phetoho's age. Substitute Azubuike's age into one of the equations to do this. In this case, the first equation is the simpler choice for this calculation.

p=6a=6(6)=36

Write your final answer: Phetoho is 36 years old and Azubuike is 6 years old.


Submit your answer as: and

Word problems: an age-old question

Chukwudi has a son, Dumile. Here are some facts about how old Chukwudi and Dumile are:

  • Chukwudi is 5 times as old as Dumile right now.
  • 5 years from now, Chukwudi will be 3 times as old as Dumile.

How old are Chukwudi and Dumile now?

Answer:

Chukwudi is years old and Dumile is years old.

numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

Start by choosing variables for the things you want to know. Then write equations with those variables to summarise the information in the question.


STEP: Choose variables for the information
[−1 point ⇒ 5 / 6 points left]

In this question we want to find the age of two people: Chukwudi and his son, Dumile. We can solve this by setting up two equations and solving them simultaneously.

Start by choosing variables to represent the ages of the father and the son. (We need to do this because we don't know the ages!) It is a good idea to choose variables that match what we are describing. So:

Let c=Chukwudi's ageLet d=Dumile's age

STEP: Write equations based on the information in the question
[−2 points ⇒ 3 / 6 points left]

Now we want to use those variables to write equations. The first piece of information from the question tells us that "Chukwudi is now 5 times as old as Dumile." As an equation, this is:

Ages now: c=5d

The second piece of information says that in "5 years... Chukwudi will be 3 times as old as his son." In 5 years Chukwudi will be c+5 years old, and similarly Dumile will be d+5 years old. Then:

Ages in 5 years: c+5=3(d+5)

STEP: Solve the equations simultaneously
[−2 points ⇒ 1 / 6 points left]

Now we have a pair of simultaneous equations! Substitute the first equation into the second equation and solve. (You can solve the equations using elimination if you prefer. In that case, the easiest option is to subtract the equations to cancel c.)

c+5=3(d+5)(5d)+5=3d+152d=10d=5

Great! Now we know that Dumile is 5 years old.


STEP: Use Dumile's age to find Chukwudi's age
[−1 point ⇒ 0 / 6 points left]

We can now find Chukwudi's age. Substitute Dumile's age into one of the equations to do this. In this case, the first equation is the simpler choice for this calculation.

c=5d=5(5)=25

Write your final answer: Chukwudi is 25 years old and Dumile is 5 years old.


Submit your answer as: and

Simultaneous equations

Solve the following equations simultaneously:

y=3x+2x=6+2y
Answer: x= and y= .
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The first thing you need to do is substitute y=3x+2 into the second equation.


STEP: Combine the equations using substitution
[−1 point ⇒ 3 / 4 points left]

We must solve the two equations 'simultaneously'. This means we want one pair of x and y values which solves both equations.

There are two methods for solving equations simultaneously: elimination and substitution. For the equations in this question, substitution is the better choice. This is because the first equation already shows us that y is equal to y=3x+2, which is exactly what we need for the substitution method. (We could solve these equations using the elimination method, but it will require more work.)

So we can substitute 3x+2 into the second equation in place of y:

x=6+2yy=3x+2x=6+2(3x+2)
NOTE: We could substitute the other variable if we want to; but the substitution above is the best choice because the y in the first equation is already the subject of the equation.

STEP: Solve for x
[−2 points ⇒ 1 / 4 points left]

Now that there is only one variable in the equation, we can solve the equation. Distribute the 2 and get on with solving the equation.

x=6+2(3x+2)x=6+2(3x)+2(2)x=66x+4x+6x=6+45x=10x=2

STEP: Solve for y
[−1 point ⇒ 0 / 4 points left]

So x=2. But remember that we also need an answer for y. It is important to remember that you can use either equation to calculate y, because the numbers we want solve both equations. It is good to pick the easiest choice. In this case, the easiest choice is the first equation, because it is arranged in a more useful way.

y=3x+2y=3(2)+2y=6+2y=4

The numbers which solve the equations simultaneously are x=2 and y=4. We can see this represented beautifully if we graph the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are x=2 and y=4.


Submit your answer as: and

Simultaneous equations

Solve the following equations simultaneously:

x=4+3yy=4x+3
Answer: x= and y= .
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The first thing you need to do is substitute y=4x+3 into the first equation.


STEP: Combine the equations using substitution
[−1 point ⇒ 3 / 4 points left]

We must solve the two equations 'simultaneously'. This means we want one pair of x and y values which solves both equations.

There are two methods for solving equations simultaneously: elimination and substitution. For the equations in this question, substitution is the better choice. This is because the second equation already shows us that y is equal to y=4x+3, which is exactly what we need for the substitution method. (We could solve these equations using the elimination method, but it will require more work.)

So we can substitute 4x+3 into the first equation in place of y:

x=4+3yy=4x+3x=4+3(4x+3)
NOTE: We could substitute the other variable if we want to; but the substitution above is the best choice because the y in the second equation is already the subject of the equation.

STEP: Solve for x
[−2 points ⇒ 1 / 4 points left]

Now that there is only one variable in the equation, we can solve the equation. Distribute the 3 and get on with solving the equation.

x=4+3(4x+3)x=4+3(4x)+3(3)x=412x+9x+12x=4+913x=13x=1

STEP: Solve for y
[−1 point ⇒ 0 / 4 points left]

So x=1. But remember that we also need an answer for y. It is important to remember that you can use either equation to calculate y, because the numbers we want solve both equations. It is good to pick the easiest choice. In this case, the easiest choice is the second equation, because it is arranged in a more useful way.

y=4x+3y=4(1)+3y=4+3y=1

The numbers which solve the equations simultaneously are x=1 and y=1. We can see this represented beautifully if we graph the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are x=1 and y=1.


Submit your answer as: and

Simultaneous equations

Solve the following equations simultaneously:

5x=205yx=2y4
Answer: x= and y= .
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The first thing you need to do is substitute x=2y4 into the first equation.


STEP: Combine the equations using substitution
[−1 point ⇒ 3 / 4 points left]

We must solve the two equations 'simultaneously'. This means we want one pair of x and y values which solves both equations.

There are two methods for solving equations simultaneously: elimination and substitution. For the equations in this question, substitution is the better choice. This is because the second equation already shows us that x is equal to x=2y4, which is exactly what we need for the substitution method. (We could solve these equations using the elimination method, but it will require more work.)

So we can substitute 2y4 into the first equation in place of x:

5x=205yx=2y45(2y4)=205y
NOTE: We could substitute the other variable if we want to; but the substitution above is the best choice because the x in the second equation is already the subject of the equation.

STEP: Solve for y
[−2 points ⇒ 1 / 4 points left]

Now that there is only one variable in the equation, we can solve the equation. Distribute the 5 and get on with solving the equation.

5(2y4)=205y5(2y)+5(4)=205y10y20=205y10y+5y=20+2015y=0y=0

STEP: Solve for x
[−1 point ⇒ 0 / 4 points left]

So y=0. But remember that we also need an answer for x. It is important to remember that you can use either equation to calculate x, because the numbers we want solve both equations. It is good to pick the easiest choice. In this case, the easiest choice is the second equation, because it is arranged in a more useful way.

x=2y4x=2(0)4x=04x=4

The numbers which solve the equations simultaneously are x=4 and y=0. We can see this represented beautifully if we graph the two equations: the answers we just found are the coordinates of the point where the lines intersect.

The values which solve these two equations are x=4 and y=0.


Submit your answer as: and

Picking a solution method

These two equations contain the same variables:

x3y=9x+5y=13
  1. These equations can be solved using either the elimination or the substitution method. Which method is better for solving these two equations?

    Answer: The better method is .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    To decide which method is better, you should look at the coefficients. For example, can the coefficients lead to the cancellation of any of the terms in these equations?


    STEP: Check the arrangement of the equations to decide which method is better
    [−1 point ⇒ 0 / 1 points left]

    This question shows us two equations. We need to decide which method is the best choice for solving the equations simultaneously. (We do not have to solve the equations!)

    Two methods we can use for solving equations simultaneously are elimination and substitution. Both of those methods can be used to solve the equations in the question. But usually one of the methods is easier than the other based on the equations. Deciding which method is better depends on the arrangement of the equations:

    • Substitution is the better choice if there is already one variable isolated or if one variable is easy to isolate. If a variable is isolated, we can substitute immediately.
    • Elimination is the better choice if there are two terms which can be cancelled easily. If the terms can cancel, we can eliminate those terms immediately.

    The equations here are:

    x3y=9x+5y=13

    In this case, the x-terms are ready to be eliminated. The coefficients of these terms are equal and opposite, which is perfect for the elimination method: it means we can cancel those two terms if we add the equations, which is how elimination works.

    The correct answer is: the elimination method.


    Submit your answer as:
  2. Solve the equations simultaneously.

    INSTRUCTION: Type your answer as a coordinate pair like this: (x;y).
    Answer: The answer is: .
    coordinate
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the method identified in Question 1.


    STEP: Eliminate (cancel) the x-terms
    [−1 point ⇒ 2 / 3 points left]

    Based on the result of Question 1, we should solve this question using elimination. It is possible to solve the equations using substitution, but using elimination is easier (as described in Question 1). Add the equations together to eliminate the x-terms. This means we add together the left and right sides of the equations.

    x3y=9equationadd thisx+5y=13xx3y+5y=9+130x+2y=42y=4


    STEP: Solve for y
    [−1 point ⇒ 1 / 3 points left]

    Now we have an equation with only one variable, and we can solve it.

    2y=4y=2

    STEP: Find the value of x
    [−1 point ⇒ 0 / 3 points left]

    So y=2. But we also need an answer for x. We can get this using the answer we just got for y: substitute this into either of the equations to get the answer. We can use either equation because we are looking for the number that solves both of them. Since we have a choice, we should pick the easier equation (if there is one). We will pick the second equation, because it has fewer negative signs.

    x+5y=13x+5(2)=13x+10=13x=3x=3

    Now we have the complete answer: the numbers which solve the equations simultaneously are x=3 and y=2.

    The correct answer is (3;2).


    Submit your answer as:

Picking a solution method

The following equations both include the variables x and y:

4x=124yy=x5
  1. These equations can be solved using either the elimination or the substitution method. Which method is better for solving these two equations?

    Answer: The better method is .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    To decide which method is better, you should look at the coefficients. For example, can the coefficients lead to the cancellation of any of the terms in these equations?


    STEP: Check the arrangement of the equations to decide which method is better
    [−1 point ⇒ 0 / 1 points left]

    This question shows us two equations. We need to decide which method is the best choice for solving the equations simultaneously. (We do not have to solve the equations!)

    Two methods we can use for solving equations simultaneously are elimination and substitution. Both of those methods can be used to solve the equations in the question. But usually one of the methods is easier than the other based on the equations. Deciding which method is better depends on the arrangement of the equations:

    • Substitution is the better choice if there is already one variable isolated or if one variable is easy to isolate. If a variable is isolated, we can substitute immediately.
    • Elimination is the better choice if there are two terms which can be cancelled easily. If the terms can cancel, we can eliminate those terms immediately.

    The equations here are:

    4x=124yy=x5

    In this case, the second equation points us to substitution: the y in that equation is isolated. This is the perfect arrangement for substitution, because we can substitute y=x5 directly into the y in the first equation.

    The correct answer is: the substitution method.


    Submit your answer as:
  2. Now find the simultaneous solution to the equations.

    INSTRUCTION: Type your answer as a coordinate pair like this: (x;y).
    Answer: The answer is: .
    coordinate
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Use the method identified in Question 1.


    STEP: Combine the equations using substitution
    [−1 point ⇒ 3 / 4 points left]

    Based on the result of Question 1, we should solve this question using substitution. It is possible to solve the equations using elimination, but using substitution is easier (as described in Question 1). Substitute y=x5 into the first equation in place of y:

    4x=124yy=x5

    Combining these equations by substution, we get:

    4x=124(x5)

    STEP: Solve for x
    [−2 points ⇒ 1 / 4 points left]

    Now we can solve the equation. Distribute the 4 and get on with solving the equation.

    4x=124(x5)4x=124(x)4(5)4x=124x+204x+4x=12+208x=8x=1

    STEP: Find the value of y
    [−1 point ⇒ 0 / 4 points left]

    So x=1. But remember that we also need an answer for y. We can find this using the answer we just got for x. It is important to remember that you can use either equation to calculate y, because the numbers we want solve both equations. It is good to pick the easiest choice. In this case, the easier choice is the second equation, because it is arranged in a more useful way.

    y=x5y=(1)5y=15y=4

    The numbers which solve the equations simultaneously are x=1 and y=4.

    The correct answer is (1;4).


    Submit your answer as:

Picking a solution method

The following equations both include the variables x and y:

9=x+2y4=4x2y
  1. These equations can be solved using either the elimination or the substitution method. Which method is better for solving these two equations?

    Answer: The better method is .
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    To decide which method is better, you should look at the coefficients. For example, can the coefficients lead to the cancellation of any of the terms in these equations?


    STEP: Check the arrangement of the equations to decide which method is better
    [−1 point ⇒ 0 / 1 points left]

    This question shows us two equations. We need to decide which method is the best choice for solving the equations simultaneously. (We do not have to solve the equations!)

    Two methods we can use for solving equations simultaneously are elimination and substitution. Both of those methods can be used to solve the equations in the question. But usually one of the methods is easier than the other based on the equations. Deciding which method is better depends on the arrangement of the equations:

    • Substitution is the better choice if there is already one variable isolated or if one variable is easy to isolate. If a variable is isolated, we can substitute immediately.
    • Elimination is the better choice if there are two terms which can be cancelled easily. If the terms can cancel, we can eliminate those terms immediately.

    The equations here are:

    9=x+2y4=4x2y

    In this case, the y-terms are ready to be eliminated. The coefficients of these terms are equal and opposite, which is perfect for the elimination method: it means we can cancel those two terms if we add the equations, which is how elimination works.

    The correct answer is: the elimination method.


    Submit your answer as:
  2. Now solve the equations simultaneously.

    INSTRUCTION: Type your answer as a coordinate pair like this: (x;y).
    Answer: The answer is: .
    coordinate
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the method identified in Question 1.


    STEP: Eliminate (cancel) the y-terms
    [−1 point ⇒ 2 / 3 points left]

    Based on the result of Question 1, we should solve this question using elimination. It is possible to solve the equations using substitution, but using elimination is easier (as described in Question 1). Add the equations together to eliminate the y-terms. This means we add together the left and right sides of the equations.

    9=x+2yequationadd this4=4x2y9+4=x4x+2y2y5=5x+0y5=5x


    STEP: Solve for x
    [−1 point ⇒ 1 / 3 points left]

    Now we have an equation with only one variable, and we can solve it.

    5=5x1=x

    STEP: Find the value of y
    [−1 point ⇒ 0 / 3 points left]

    So x=1. But we also need an answer for y. We can get this using the answer we just got for x: substitute this into either of the equations to get the answer. We can use either equation because we are looking for the number that solves both of them. Since we have a choice, we should pick the easier equation (if there is one). In this case, neither of the equations is much more friendly than the other, so we will just use the first equation.

    9=x+2y9=(1)+2y9=1+2y8=2y4=y

    Now we have the complete answer: the numbers which solve the equations simultaneously are x=1 and y=4.

    The correct answer is (1;4).


    Submit your answer as:

Setting up simultaneous equations

Last week, Adaeze and Babalwe had a maths test. Now they are comparing their marks and they notice these facts:

  • The sum of the marks is 133.
  • Adaeze's mark is 11 more than Babalwe's mark.

Let a represent Adaeze's mark and b represent Babalwe's mark. Then which equations below accurately represent the facts? Select your answer from the choices below.

Answer:
Fact about the test scores Equation
The sum of the marks is 133.
Adaeze's mark is 11 more than Babalwe's mark.
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

For the first equation, the key word is sum. For the second equation the key word is more. Use these key words to change the words into operations, for example, addition or multiplication.


STEP: Translate the first fact into an equation
[−1 point ⇒ 1 / 2 points left]

In this question we need to translate words into equations. This can be challenging. One useful approach is to look for important words which tell us what numbers and calculations to use. Here are some common words and what they mean when we write mathematical expressions and equations:

Word Meaning
sum +
product ×
is =
consecutive 1 apart
more than add to
less than subtract from

With these key words in mind, let's identify the key parts/words in each of these facts. Then we can translate each of the parts into maths.

The question tells us that we should use the variable a for Adaeze's mark and b for Babalwe's mark. So we can break up the first fact like this:

The sum of the marksis133a+b=133

The first fact is equivalent to this equation: a+b=133.


STEP: Translate the second fact into an equation
[−1 point ⇒ 0 / 2 points left]

Similarly, we can identify key parts of the second fact, and translate each into an expression.

Adaeze's markis11 more than Babalwe's marka=b+11

This equation, a=b+11, means that Adaeze's mark is more that Babalwe's mark.

The correct answers are:

Fact about the test scores Equation
The sum of the marks is 133. a+b=133
Adaeze's mark is 11 more than Babalwe's mark. a=b+11

Submit your answer as: and

Setting up simultaneous equations

Last week, Adefoluke and Bukelwa had a chemistry test. Now they are comparing their marks and they notice these facts:

  • The sum of the marks is 121.
  • Adefoluke's mark is 9 more than Bukelwa's mark.

Let a represent Adefoluke's mark and b represent Bukelwa's mark. Then which equations below accurately represent the facts? Select your answer from the choices below.

Answer:
Fact about the test scores Equation
The sum of the marks is 121.
Adefoluke's mark is 9 more than Bukelwa's mark.
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

For the first equation, the key word is sum. For the second equation the key word is more. Use these key words to change the words into operations, for example, addition or multiplication.


STEP: Translate the first fact into an equation
[−1 point ⇒ 1 / 2 points left]

In this question we need to translate words into equations. This can be challenging. One useful approach is to look for important words which tell us what numbers and calculations to use. Here are some common words and what they mean when we write mathematical expressions and equations:

Word Meaning
sum +
product ×
is =
consecutive 1 apart
more than add to
less than subtract from

With these key words in mind, let's identify the key parts/words in each of these facts. Then we can translate each of the parts into maths.

The question tells us that we should use the variable a for Adefoluke's mark and b for Bukelwa's mark. So we can break up the first fact like this:

The sum of the marksis121a+b=121

The first fact is equivalent to this equation: a+b=121.


STEP: Translate the second fact into an equation
[−1 point ⇒ 0 / 2 points left]

Similarly, we can identify key parts of the second fact, and translate each into an expression.

Adefoluke's markis9 more than Bukelwa's marka=b+9

This equation, a=b+9, means that Adefoluke's mark is more that Bukelwa's mark.

The correct answers are:

Fact about the test scores Equation
The sum of the marks is 121. a+b=121
Adefoluke's mark is 9 more than Bukelwa's mark. a=b+9

Submit your answer as: and

Setting up simultaneous equations

Last week, Andrew and Babatunde had a physics test. Now they are comparing their marks and they notice these facts:

  • The sum of the marks is 153.
  • Andrew's mark is 11 more than Babatunde's mark.

Let a represent Andrew's mark and b represent Babatunde's mark. Then which equations below accurately represent the facts? Select your answer from the choices below.

Answer:
Fact about the test scores Equation
The sum of the marks is 153.
Andrew's mark is 11 more than Babatunde's mark.
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

For the first equation, the key word is sum. For the second equation the key word is more. Use these key words to change the words into operations, for example, addition or multiplication.


STEP: Translate the first fact into an equation
[−1 point ⇒ 1 / 2 points left]

In this question we need to translate words into equations. This can be challenging. One useful approach is to look for important words which tell us what numbers and calculations to use. Here are some common words and what they mean when we write mathematical expressions and equations:

Word Meaning
sum +
product ×
is =
consecutive 1 apart
more than add to
less than subtract from

With these key words in mind, let's identify the key parts/words in each of these facts. Then we can translate each of the parts into maths.

The question tells us that we should use the variable a for Andrew's mark and b for Babatunde's mark. So we can break up the first fact like this:

The sum of the marksis153a+b=153

The first fact is equivalent to this equation: a+b=153.


STEP: Translate the second fact into an equation
[−1 point ⇒ 0 / 2 points left]

Similarly, we can identify key parts of the second fact, and translate each into an expression.

Andrew's markis11 more than Babatunde's marka=b+11

This equation, a=b+11, means that Andrew's mark is more that Babatunde's mark.

The correct answers are:

Fact about the test scores Equation
The sum of the marks is 153. a+b=153
Andrew's mark is 11 more than Babatunde's mark. a=b+11

Submit your answer as: and